if a runner races 50 meters in 5 seconds, how fast is she going?

Answers

Answer 1
The answer is she is going 10 meters a second
Answer 2

Answer:

10 m/s

Step-by-step explanation:

The phrase "how fast she is going" tells us that we need to find her speed.

To find her speed, we need to take her distance (50 meters) and divide it by the time (5 seconds):

Runner's Speed = Distance ÷ Time

Runner's Speed = 50 ÷ 5

Runner's Speed = 10 m/s

Hence, the girl's speed is 10 m/s


Related Questions

The Dubois formula relates a person's surface area s
(square meters) to weight in w (kg) and height h
(cm) by s =0.01w^(1/4)h^(3/4). A 60kg person is
150cm tall. If his height doesn't change but his w

Answers

The Dubois formula relates: The surface area of the person is increasing at a rate of approximately 0.102 square meters per year when his weight increases from 60kg to 62kg.

Given:

s = 0.01w^(1/4)h^(3/4) (Dubois formula)

w1 = 60kg (initial weight)

w2 = 62kg (final weight)

h = 150cm (constant height)

To find the rate of change of surface area with respect to weight, we can differentiate the Dubois formula with respect to weight and then substitute the given values:

ds/dw = (0.01 × (1/4) × w^(-3/4) × h^(3/4)) (differentiating the formula with respect to weight)

ds/dw = 0.0025 × h^(3/4) × w^(-3/4) (simplifying)

Substituting the values w = 60kg and h = 150cm, we can calculate the rate of change:

ds/dw = 0.0025 × (150cm)^(3/4) × (60kg)^(-3/4)

ds/dw ≈ 0.102 square meters per kilogram

Therefore, when the person's weight increases from 60kg to 62kg, his surface area is increasing at a rate of approximately 0.102 square meters per year.

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Complete question:
The Dubois formula relates a person's surface area s (square meters) to weight in w (kg) and height h (cm) by s =0.01w^(1/4)h^(3/4). A 60kg person is 150cm tall. If his height doesn't change but his weight increases by 0.5kg/yr, how fast is his surface area increasing when he weighs 62kg?

In order to set rates, an insurance company is trying to estimate the number of sick days that full time workers at an auto repair shop take per yearA previous selected if the company wants to be 95% confident that the true mean differs from the sample mean by no more than 1 day? OA 31 OB. 141 OC. 1024 OD. 512 nys that full time workerslat an auto repair shop take per year A previous study indicated that the population staridard deviation is 2.8 days How turpe a sampio must do e sample mean by no more than 1 day?

Answers

The insurance company would need to take a sample of 31 full-time workers from the auto repair shop to estimate the population mean with a margin of error no more than 1 day at a 95% confidence level.

To estimate the number of sick days that full-time workers at an auto repair shop take per year, the insurance company needs to take a sample from the population of workers at the shop. The sample size required to estimate the population mean with a margin of error of no more than 1 day can be calculated using the formula:

n = (z^2 * σ²) / E²

where:
z = the z-score corresponding to the desired level of confidence (in this case, 95% confidence corresponds to z = 1.96)
σ = the population standard deviation (given as 2.8 days)
E = the maximum allowable margin of error (given as 1 day)

Plugging in the values, we get:

n = (1.96² * 2.8^2) / 1²
n ≈ 31

Therefore, the insurance company would need to take a sample of 31 full-time workers from the auto repair shop to estimate the population mean with a margin of error no more than 1 day at a 95% confidence level.

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a probability model include P yellow = 2/9 and P black = 5/18 select all probabilities that could complete the model

P white = 2/9 P orange = 5/9

P white = 1/6 P orange = 1/3

P white = 2/7 P orange = 2/7

P white = 1/10 P orange = 2/5

P white = 2/9 P orange = 1/9

Answers

The probabilities that could complete the model in this problem are given as follows:

P white = 2/9 P orange = 5/9P white = 1/6 P orange = 1/3.

How to calculate a probability?

The parameters that are needed to calculate a probability are listed as follows:

Number of desired outcomes in the context of a problem or experiment.Number of total outcomes in the context of a problem or experiment.

Then the probability is then calculated as the division of the number of desired outcomes by the number of total outcomes.

For a valid probability model, the sum of all the probabilities in the model must be of one.

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Eight Tires Of Different Brands Are Ranked From 1 To 8 (Best To Worst) According To Mileage Performance. Suppose Four Of These Tires Are Chosen At Random By A Customer. Let Y Denote The Actual Quality Rank Of The Best Tire Selected By The Customer. Find The Probabilities Associated With All Of The Possible Values Of Y. (Enter Your Probabilities As

Answers

The probabilities associated with all possible values of Y are:

P(Y = 1) = 1/2

P(Y = 2) = 1/2

P(Y = 3) = 1/2

P(Y = 4) = 1/8

To find the probabilities associated with all possible values of Y, consider the different scenarios of tire selection.

Since there are eight tires and four are chosen at random, the possible values of Y range from 1 to 4.

1. Y = 1 (The best tire is selected)

  In this case, the best tire can be selected in any of the four positions (1st, 2nd, 3rd, or 4th). The remaining three tires can be any of the remaining seven tires. Therefore, the probability is:

  P(Y = 1) = (4/8) * (7/7) * (6/6) * (5/5) = 1/2

2. Y = 2 (The second-best tire is selected)

  In this case, the second-best tire can be selected in any of the four positions (1st, 2nd, 3rd, or 4th). The best tire is not selected, so it can be any of the remaining seven tires. The remaining two tires can be any of the remaining six tires. Therefore, the probability is:

  P(Y = 2) = (4/8) * (7/7) * (6/6) * (5/5) = 1/2

3. Y = 3 (The third-best tire is selected)

  In this case, the third-best tire can be selected in any of the four positions (1st, 2nd, 3rd, or 4th). The best tire is not selected, so it can be any of the remaining seven tires. The second-best tire is also not selected, so it can be any of the remaining six tires. The remaining tire can be any of the remaining five tires. Therefore, the probability is:

  P(Y = 3) = (4/8) * (7/7) * (6/6) * (5/5) = 1/2

4. Y = 4 (The fourth-best tire is selected)

  In this case, the fourth-best tire is selected in the only position left. The best tire is not selected, so it can be any of the remaining seven tires. The second-best and third-best tires are also not selected, so they can be any of the remaining six tires. Therefore, the probability is:

  P(Y = 4) = (1/8) * (7/7) * (6/6) * (5/5) = 1/8

In summary, the probabilities associated with all possible values of Y are:

P(Y = 1) = 1/2

P(Y = 2) = 1/2

P(Y = 3) = 1/2

P(Y = 4) = 1/8

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5) You have money in an account at 6% interest, compounded quarterly. To the nearest year, how long will it take for your money to double? A) 12 years D) 7 years B) 9 years C) 16 years

Answers

The nearest year it will take for your money to double at a 6% interest compounded quarterly is 12 years.

If you have money in an account at 6% interest, compounded quarterly and you want to know how long it will take for your money to double, you can use the formula for compound interest: A = P [tex](1 + r/n)^{(nt)}[/tex] Where: A = the final amount of money after t years = the principal (initial) amount of money = the annual interest rate = the number of times the interest is compounded per year = the number of years it is invested this problem, we are looking for when A = 2P since that is when the money has doubled. So we can set up the equation:2P = P (1 + 0.06/4)^(4t)Simplifying:2 =[tex](1 + 0.015)^{4t}[/tex] Taking the logarithm of both sides to solve for t: ln 2 = ln [tex](1.015)^{(4t)}[/tex] Using the property of logarithms that ln [tex]a^b[/tex] = b ln a: ln 2 = 4t ln (1.015)Dividing both sides by 4 ln (1.015):t = ln 2 / (4 ln (1.015))t ≈ 11.896 Rounding to the nearest year: t ≈ 12, so it will take about 12 years for the money to double. Therefore, the correct answer is A) 12 years.

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Consider the curves y = 3x2 +6x and y = -42 +4. a) Determine their points of intersection (1.01) and (22,92)ordering them such that 1

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The problem asks us to find the points of intersection between two curves, y = 3x^2 + 6x and y = -4x^2 + 42. The given points of intersection are (1.01) and (22, 92), and we need to order them such that the x-values are in ascending order.

To find the points of intersection, we set the two equations equal to each other and solve for x: 3x^2 + 6x = -4x^2 + 42. Simplifying the equation, we get 7x^2 + 6x - 42 = 0. Solving this quadratic equation, we find two solutions: x ≈ -3.21 and x ≈ 1.01. Given the points of intersection (1.01) and (22, 92), we order them in ascending order of their x-values: (-3.21, -42) and (1.01, 10.07). Therefore, the ordered points of intersection are (-3.21, -42) and (1.01, 10.07).

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2. Use an integral to find the area above the curve y=-e* + e(2x-3) and below the x-axis, for x 20. You need to use a graph to answer this question. You will not receive any credit if you use the meth

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To find the area above the curve y = -e^x + e^(2x-3) and below the x-axis for x ≥ 0, we can use an integral. The area can be calculated by integrating the absolute value of the function from the point where it intersects the x-axis to infinity.

Let's denote the given function as f(x) = -e^x + e^(2x-3). We want to find the integral of |f(x)| with respect to x from the x-coordinate where f(x) intersects the x-axis to infinity.

First, we need to find the x-coordinate where f(x) intersects the x-axis. Setting f(x) = 0, we have:

-e^x + e^(2x-3) = 0

Simplifying the equation, we get:

e^x = e^(2x-3)

Taking the natural logarithm of both sides, we have:

x = 2x - 3

Solving for x, we find x = 3.

Now, the integral for the area can be written as:

A = ∫[3, ∞] |f(x)| dx

Substituting the expression for f(x), we have:

A = ∫[3, ∞] |-e^x + e^(2x-3)| dx

By evaluating this integral using appropriate techniques, such as integration by substitution or integration by parts, we can find the exact value of the area.

Please note that a graph of the function is necessary to visualize the region and determine the bounds of integration accurately.

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Use Mathematical Induction to show that that the solution to the recurrence relation T (n) = aT ( [7]) with base condition T(1) = c is T(n) = callogn 27

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By induction, we have shown that if the formula holds for k, then it also holds for k+1. Since it holds for the base case T(1) = c, we can conclude that the formula T(n) = c * (a log₇ n) is the solution to the given recurrence relation T(n) = aT(n/7) with base condition T(1) = c.

Paragraph 1: The solution to the recurrence relation T(n) = aT(n/7) with base condition T(1) = c is given by T(n) = c * (a log₇ n), where c and a are constants. This formula represents the closed-form solution for the recurrence relation and is derived using mathematical induction.

Paragraph 2: We begin the proof by showing that the formula holds for the base case T(1) = c. Substituting n = 1 into the formula, we get T(1) = c * (a log₇ 1) = c * 0 = c, which matches the given base condition.

Next, we assume that the formula holds for some positive integer k, i.e., T(k) = c * (a log₇ k). Now, we need to prove that it also holds for the next value, k+1. Substituting n = k+1 into the recurrence relation, we have T(k+1) = aT((k+1)/7). Using the assumption, we can rewrite this as T(k+1) = a * (c * (a log₇ (k+1)/7)). Simplifying further, we get T(k+1) = c * (a log₇ (k+1)).

By induction, we have shown that if the formula holds for k, then it also holds for k+1. Since it holds for the base case T(1) = c, we can conclude that the formula T(n) = c * (a log₇ n) is the solution to the given recurrence relation T(n) = aT(n/7) with base condition T(1) = c.

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00 = Use the power series = (-1)"x" to determine a power series 1+x representation, centered at 0, for the given function, f(x) = ln(1 + 3x?). n=0 =

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The power series representation, centered at 0, for the function f(x) = ln(1 + 3x), using the power series (-1)ⁿx, is ∑(-1)ⁿ(3x)ⁿ/n, where n ranges from 0 to infinity.

To find the power series representation of ln(1 + 3x) centered at 0, we can use the formula for the power series expansion of ln(1 + x):

ln(1 + x) = ∑(-1)ⁿ(xⁿ/n)

In this case, we have 3x instead of just x, so we replace x with 3x:

ln(1 + 3x) = ∑(-1)ⁿ((3x)ⁿ/n)

Now, we can rewrite the series using the power series (-1)ⁿx:

ln(1 + 3x) = ∑(-1)ⁿ(3x)ⁿ/n

This is the power series representation, centered at 0, for the function ln(1 + 3x) using the power series (-1)ⁿx. The series starts with n = 0 and continues to infinity.

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Find z such that 62.1% of the standard normal curve lies to the left of z. a. –0.308 b. 0.494 c. 0.308 d. –1.167 e. 1.167

Answers

normal curve lies to the left of option c. 0.308.

To find the value of z such that 62.1% of the standard normal curve lies to the left of z, we need to use the standard normal distribution table or a statistical calculator.

Using a standard normal distribution table or a calculator, we can find the z-value associated with the cumulative probability of 62.1%. The closest value in the standard normal distribution table to 62.1% is 0.6116.

The z-value associated with a cumulative probability of 0.6116 is approximately 0.308.

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(5 points) Find the arclength of the curve r(t) = (6 sint, -6, 6 cost), -8

Answers

The arclength of the curve is given by 6t + 48.

The given curve is r(t) = (6 sint, -6, 6 cost), -8.

The formula for finding the arclength of the curve is shown below:

S = ∫├ r'(t) ├ dt Here, r'(t) is the derivative of r(t).

For the given curve, r(t) = (6sint, -6, 6cost)

So, we need to find r'(t)

First, differentiate each component of r(t) w.r.t t.r'(t) = (6cost, 0, -6sint)

Simplifying the above expression gives us│r'(t) │= √(6²cos²t + (-6sin t)²)│r'(t) │

= √(36 cos²[tex]-8t^{t}[/tex] + 36 sin²t)│r'(t) │

= 6So the arclength of the curve is

S = ∫├ r'(t) ├ dt

= ∫6dt [lower limit

= -8, upper limit

= t]S = [6t] |_ -8^t

= 6t - (-48)S = 6t + 48

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(1 point) Calculate the derivative. d sele ſi sec( 4r + 19) de dt J87 sec(4t+19) On what interval is the derivative defined?

Answers

The chain rule can be used to determine the derivative of the given function. The function should be written as y = sec(4t + 19).

We discriminate y with regard to t using the chain rule:

Dy/dt = Dy/Du * Dy/Dt

It has u = 4t + 19.Let's discover dy/du first. Sec(u)'s derivative with regard to u is given by:

Sec(u) * Tan(u) = d(sec(u))/du.Let's locate du/dt next. Simply 4, then, is the derivative of u = 4t + 19 with regard to t.We can now reintroduce these derivatives into the chain rule formula as follows:dy/dt is equal to dy/du * du/dt, which is equal to sec(u) * tan(u) * 4 = 4sec(u) * tan(u).

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9. (15 points) Evaluate the integral √4-7 +√4-2³-y (x² + y² +22)³/2dzdydz

Answers

The value of the integral is given as 5225/32 (14π/3 + 8), which is the answer to the problem.

The given integral to be evaluated is:

∫∫∫[√(4 - 7 + x² + y²) + √(4 - 2³ - y)][(x² + y² + 22)³/2] dz dy dx or, ∫∫∫[√(x² + y² - 3) + √(1 - y)][(x² + y² + 22)³/2] dz dy dx

Now, let's compute the integral using cylindrical coordinates.

The conversion formula from cylindrical coordinates to rectangular coordinates is:

x = r cos θ, y = r sin θ and z = z

Hence, the given integral is:

∫∫∫[√(r² - 3) + √(1 - r sin θ)][r³(cos²θ + sin²θ + 22)³/2] rdz dr dθ

Bounds of the integral:

z: 0 to √(3 - r²) and r: 1 to √3 and θ: 0 to 2π∫₀²π ∫₁ᵣ √3 ∫₀^√(3-r²) [√(r² - 3) + √(1 - r sin θ)][r³(cos²θ + sin²θ + 22)³/2] dz dr dθ

We can evaluate the integral by performing the following substitutions:

Let u = 3 - r² → du = -2rdr

Let v = rsinθ → dv = rcosθdθ

Now, the integral becomes:

∫₀²π ∫₀¹ ∫₀√(3-r²) [√(r² - 3) + √(1 - v)][(r² + v² + 22)³/2] rdv du dθ

Using the partial fraction method, we can evaluate the second integral:

∫₀²π ∫₀¹ [1/2(√r² - 3 - √(1 - v))] + [(r² + v² + 22)³/2] dv du dθ

For the first integral, let's make a substitution, u = r² - 3; this implies du = 2r dr.∫₀²π ∫₀¹ [1/2(√u - √(1 - v))] + [(u + v² + 25)³/2] dv du dθ

On solving, the value of the integral is given as 5225/32 (14π/3 + 8), which is the answer to the problem.

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ANSWER CORRECTLY AND PROVIDE A DETAILED SOLUTION.
TOPIC: HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS.
2. (D³ - D²4)y = 0

Answers

The general solution to the homogenous linear differential equation (D³ - D²4)y = 0 is given by y = C₁ + C₂e^(2t) + C₃e^(-2t), where C₁, C₂, and C₃ are arbitrary constants.

To explain the process in more detail, let's start by considering the differential equation (D³ - D²4)y = 0, where D represents the derivative operator with respect to t. To solve this equation, we introduce the characteristic equation by replacing D with lambda, yielding (lambda³ - lambda²4) = 0.

Now, we solve the characteristic equation to find its roots. Factoring out lambda, we have lambda²(lambda - 4) = 0. This equation is satisfied when lambda = 0 and when lambda - 4 = 0, leading to two additional roots: lambda = 0 and lambda = ±2.

Based on the roots of the characteristic equation, we can write the general solution to the differential equation. The general solution takes the form y = C₁e^(0t) + C₂e^(2t) + C₃e^(-2t), where C₁, C₂, and C₃ are arbitrary constants.

The term e^(0t) simplifies to e^0, which is equal to 1. Thus, the first term in the general solution becomes C₁.

For the terms e^(2t) and e^(-2t), we keep the exponential functions intact, as they represent linearly independent solutions. The coefficients C₂ and C₃ allow for different combinations of these solutions.

Therefore, the general solution to the homogenous linear differential equation (D³ - D²4)y = 0 is given by y = C₁ + C₂e^(2t) + C₃e^(-2t), where C₁, C₂, and C₃ are arbitrary constants.

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Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = [1, 2, -2] b = [6, 0, -8] exact o approximate

Answers

The angle between vectors a and b is approximately 44 degrees.

What is vector?

A vector is a quantity that not only indicates magnitude but also indicates how an object is moving or where it is in relation to another point or item.

To find the angle between two vectors, you can use the dot product formula:

a · b = |a| |b| cos(θ)

where a · b is the dot product of vectors a and b, |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them.

Let's calculate the dot product first:

a · b = (1)(6) + (2)(0) + (-2)(-8)

     = 6 + 0 + 16

     = 22

Next, we calculate the magnitudes of the vectors:

|a| = √(1^2 + 2^2 + (-2)^2) = √(1 + 4 + 4) = √9 = 3

|b| = √(6^2 + 0^2 + (-8)^2) = √(36 + 0 + 64) = √100 = 10

Now, substituting the values into the dot product formula:

22 = (3)(10) cos(θ)

Dividing both sides by 30:

22/30 = cos(θ)

Taking the inverse cosine [tex](cos^{-1})[/tex] of both sides to solve for θ:

[tex]\theta = cos^{-1}(22/30)[/tex]

Now, let's calculate the angle using an exact expression:

[tex]\theta = cos^{-1}(22/30)[/tex] ≈ 0.7754 radians

To approximate the angle to the nearest degree, we convert radians to degrees:

θ ≈ 0.7754 × (180/π) ≈ 44.4 degrees

Therefore, the angle between vectors a and b is approximately 44 degrees.

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simplify 8-(root)112 all over 4

Answers

Answer:

2 - √7 ≈  -0.64575131

Step-by-step explanation:

simplify  (8 - √112)/4

√112 = √(16 * 7) = √16 * √7 = 4√7

substitute

(8 - √112)/4 = (8 - 4√7)/4

simplify the numerator by dividing each term by 4:

8/4 - (4√7)/4 = 2 - √7/1

write the simplified expression as:

2 - √7 ≈  -0.64575131

"Evaluate the indefinite Integral. x/1+x4 dx

Answers

To evaluate the indefinite integral of the function f(x) = x/(1 + x^4) dx, we can use the method of partial fractions. Here's the step-by-step process:

1. Start by factoring the denominator: 1 + x^4. We can rewrite it as (1 + x^2)(1 - x^2).

2. Express the fraction x/(1 + x^4) in terms of partial fractions. We'll need to find the constants A, B, C, and D to represent the partial fractions:

  x/(1 + x^4) = A/(1 + x^2) + B/(1 - x^2)

3. Clear the fractions by multiplying both sides of the equation by (1 + x^4):

  x = A(1 - x^2) + B(1 + x^2)

4. Expand and collect like terms:

  x = A - Ax^2 + B + Bx^2

5. Equate the coefficients of like powers of x:

  -Ax^2 + Bx^2 = 0x^2

  A + B = 1

6. From the equation -Ax^2 + Bx^2 = 0x^2, we can conclude that A = B. Substituting this into A + B = 1:

  A + A = 1

  2A = 1

  A = 1/2

  B = A = 1/2

7. Now we can rewrite the original fraction using the values of A and B:

  x/(1 + x^4) = 1/2(1/(1 + x^2) + 1/(1 - x^2))

8. The integral becomes:

  ∫(x/(1 + x^4)) dx = ∫(1/2(1/(1 + x^2) + 1/(1 - x^2))) dx

9. Split the integral into two parts:

  ∫(1/2(1/(1 + x^2) + 1/(1 - x^2))) dx = 1/2(∫(1/(1 + x^2)) dx + ∫(1/(1 - x^2)) dx)

10. Evaluate the integrals:

  ∫(1/(1 + x^2)) dx = arctan(x) + C1

  ∫(1/(1 - x^2)) dx = 1/2ln|((1 + x)/(1 - x))| + C2

11. Combining the results, we get:

  ∫(x/(1 + x^4)) dx = 1/2(arctan(x) + 1/2ln|((1 + x)/(1 - x))|) + C

So, the indefinite integral of x/(1 + x^4) dx is 1/2(arctan(x) + 1/2ln|((1 + x)/(1 - xx))|) + C, where C is the constant of integration.

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If z = x2 − xy 5y2 and (x, y) changes from (3, −1) to (3. 03, −1. 05), compare the values of δz and dz. (round your answers to four decimal places. )

Answers

If z = x2 − xy 5y2 and (x, y) changes from (3, −1) to (3. 03, −1. 05), the values of δz and dz when (x, y) change from (3, −1) to (3.03, −1.05) are -2.1926 and 0.63 respectively.

As we know,  z = x² - xy - 5y². We have to find the comparison between δz and dz when (x, y) changes from (3, −1) to (3.03, −1.05). The total differential of z, dz IS:

dz = ∂z/∂x dx + ∂z/∂y dyδz = z(3.03, -1.05) - z(3, -1)

The partial derivatives of z with respect to x and y can be calculated as:

∂z/∂x = 2x - y∂z/∂y = -x - 10y

Let (x, y) change from (3, −1) to (3.03, −1.05).

Then change in x, δx = 3.03 - 3 = 0.03

Change in y, δy = -1.05 - (-1) = -0.05

δz = z(3.03, -1.05) - z(3, -1)

δz = (3.03)² - (3.03)(-1) - 5(-1.05)² - [3² - 3(-1) - 5(-1)²]

δz = 9.1809 + 3.09 - 5.5125 - 8.95δz = -2.1926

Round δz to four decimal places,δz = -2.1926

dz = ∂z/∂x

δx + ∂z/∂y δydz = (2x - y) dx - (x + 10y) dy

When (x, y) = (3, -1), we have,

dz = (2(3) - (-1)) (0.03) - ((3) + 10(-1))(-0.05)

dz = (6 + 0.03) - (-7) (-0.05)

dz ≈ 0.63

Round dz to four decimal places, dz ≈ 0.63

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Optimization Suppose an airline policy states that all baggage must be box-shaped, with a square base. Additionally, the sum of the length, width, and height must not exceed 126 inches. Write a functio to represent the volume of such a box, and use it to find the dimensions of the box that will maximize its volume. Length = inches 1 I Width = inches Height = inches

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The volume of a box-shaped baggage with a square base can be represented by the function V(l, w, h) = l^2 * h. To find the dimensions that maximize the volume, we need to find the critical points of the function by taking its partial derivatives with respect to each variable and setting them to zero.

Let's denote the length, width, and height as l, w, and h, respectively. We are given that l + w + h ≤ 126. Since the base is square-shaped, l = w.

The volume function becomes V(l, h) = l^2 * h. Substituting l = w, we get V(l, h) = l^2 * h.

To find the critical points, we differentiate the volume function with respect to l and h:

dV/dl = 2lh

dV/dh = l^2

Setting both derivatives to zero, we have 2lh = 0 and l^2 = 0. Since l > 0, the only critical point is at l = 0.

However, the constraint l + w + h ≤ 126 implies that l, w, and h must be positive and nonzero. Therefore, the dimensions that maximize the volume cannot be determined based on the given constraint.

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The lengths of two sides of a triangle are 2x² - 10x + 6 inches and x²-x-4 inches. If the perimeter of the triangle is 3x² - 7x + 2 inches, find the length of the third side.
[Hint: draw and label a picture]​

Answers

Answer:

Length of third side = 4x inches

Step-by-step explanation:

The perimeter of a triangle is the sum of the lengths of its three sides.

Step 1:  First we need to add the two sides we have and simplify:

2x^2 - 10x + 6 + x^2 - x - 4

(2x^2 + x^2) + (-10x - x) + (6 - 4)

3x^2 - 11x + 2

Step 2:  Now, we need to subtract this from the perimeter to find the length of the third side:

Third side = 3x^2 - 7x + 2 - (3x^2 - 11x + 2)

Third side = 3x^2 - 7x + 2 - 3x^2 + 11x - 2

Third side = 4x

Thus, the length of the third side is 4x inches

Optional Step 3:  We can check the validity of our answer by seeing if the sum of the lengths of the three sides equals the perimeter we're given

3x^2 - 7x + 2 = (2x^2 - 10x + 6) + (x^2 - x - 4) + (4x)

3x^2 - 7x + 2 = (2x^2 + x^2) + (-10x - x + 4x) + (6 - 4)

3x^2 - 7x + 2 = 3x^2 + (-11x + 4x) + 2

3x^2 - 7x + 2 = 3x^2 - 7x + 2

Thus, we've correctly found the length of the third side.

I attached a picture of a triangle that shows the info we're given and the answer we found.

Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t)=v'(t)=g, where g= -9.8 m/s? a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. A softball is popped up vertically (from the ground) with a velocity of 33 m/s. a. v(t) = 1 b. s(t)= c. The object's highest point is m at time t=s. (Simplify your answers. Round to two decimal places as needed.) d.to (Simplify your answer. Round to two decimal places as needed.)

Answers

The calculations involve finding  vertical motion of an object subject to gravity and position of the object at different times, determining the time at the highest point, and finding the time of impact with the ground.

What are the calculations and information needed to determine the vertical motion of an object subject to gravity?

In the given scenario, the object is experiencing vertical motion due to gravity. We are required to find the velocity, position, time at the highest point, and time when it strikes the ground.

a. To find the velocity at any time, we integrate the acceleration equation, yielding v(t) = -9.8t + C, where C is the constant of integration.

b. The position can be found by integrating the velocity equation, giving s(t) = -4.9t^2 + Ct + D, where D is another constant of integration.

c. To find the time at the highest point, we set the velocity equation equal to zero and solve for t. The height at this point is given by substituting the obtained time into the position equation.

d. To find the time when the object strikes the ground, we set the position equation equal to zero and solve for t.

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a) Suppose ^ is an eigenvalue of A, i.e. there is a vector v such that Av = Iv. Show that cA + d is an
eigenvalue of B = cA + dI. Hint: Compute Bv.
b) Suppose A is an eigenvalue of A. Argue that 12 is an eigenvalue of A2.

Answers

a) Bv = (^c + d)v.  b)  v is an eigenvector of A2 with eigenvalue [tex]A^3[/tex]. Thus, 12 is an eigenvalue of A2, if A is an eigenvalue of A.

a) Let us assume that ^ is an eigenvalue of A and let v be the eigenvector corresponding to it.

Then, Av = ^v

Now, we need to find if cA + d is an eigenvalue of B. We have, B = cA + dI andBv = (cA + dI)v = cAv + dvNow, we can substitute Av from the above equation to get

Bv = cAv + dv = c(^v) + dv= ^cv + dv = (^c + d)v

Hence,

which shows that cA + d is indeed an eigenvalue of B, with eigenvector v.

b) Let us assume that A is an eigenvalue of A, with eigenvector v corresponding to it. Then, Av = Av^2 = AAv= A^2v

Now, we need to find the eigenvalue corresponding to the eigenvector v of A2. We have,

A2v = AA.v = A([tex]A^2[/tex]v)

Substituting A^2v from above, we get

A2v = A([tex]A^2[/tex]v) = [tex]A^3[/tex]v

Hence, v is an eigenvector of A2 with eigenvalue [tex]A^3[/tex]. Thus, 12 is an eigenvalue of A2, if A is an eigenvalue of A.

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differential equations
Solve general solution of the #F: (D² - 2D³ -2D² -3D-2) + =0 Ym-Y = 4-3x² (D² +1) + = 12 cos²x DE

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the general solution of the differential equation as y = y_c + y_p. This general solution accounts for both the homogeneous and non-homogeneous terms in the original equation.

The given differential equation is (D² - 2D³ - 2D² - 3D - 2)y = 4 - 3x²(D² + 1) + 12cos²(x).

To find the general solution, we first need to find the complementary solution by solving the homogeneous equation (D² - 2D³ - 2D² - 3D - 2)y = 0. This equation can be factored as (D + 2)(D + 1)(D² - 2D - 1)y = 0.

The characteristic equation associated with the homogeneous equation is (r + 2)(r + 1)(r² - 2r - 1) = 0. Solving this equation gives us the roots r1 = -2, r2 = -1, r3 = 1 + √2, and r4 = 1 - √2.

The complementary solution is given by y_c = c1e^(-2x) + c2e^(-x) + c3e^((1 + √2)x) + c4e^((1 - √2)x), where c1, c2, c3, and c4 are arbitrary constants.

Next, we need to find the particular solution based on the non-homogeneous terms. For the term 4 - 3x²(D² + 1), we assume a particular solution of the form y_p = a + bx + cx² + dcos(x) + esin(x), where a, b, c, d, and e are coefficients to be determined.

By substituting y_p into the differential equation, we can determine the values of the coefficients. Equating coefficients of like terms, we can solve for a, b, c, d, and e.

Finally, combining the complementary and particular solutions, we obtain the general solution of the differential equation as y = y_c + y_p. This general solution accounts for both the homogeneous and non-homogeneous terms in the original equation.

Note: The exact coefficients and form of the particular solution will depend on the specific values and terms given in the original equation, as well as the methods used to find the coefficients.

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A production line is equipped with two quality control check points that tests all items on the line. At check point =1, 10% of all items failed the test. At check point =2, 12% of all items failed the test. We also know that 3% of all items failed both tests. A. If an item failed at check point #1, what is the probability that it also failed at check point #22 B. If an item failed at check point #2, what is the probability that it also failed at check point =12 C. What is the probability that an item failed at check point #1 or at check point #2? D. What is the probability that an item failed at neither of the check points ?

Answers

The probabilities as follows:

A. P(F2|F1) = 0.3 (30%)

B. P(F1|F2) = 0.25 (25%)

C. P(F1 or F2) = 0.19 (19%)

D. P(not F1 and not F2) = 0.81 (81%)

To solve this problem, we can use the concept of conditional probability and the principle of inclusion-exclusion.

Given:

P(F1) = 0.10 (Probability of failing at Check Point 1)

P(F2) = 0.12 (Probability of failing at Check Point 2)

P(F1 and F2) = 0.03 (Probability of failing at both Check Point 1 and Check Point 2)

A. To find the probability that an item failed at Check Point 1 and also failed at Check Point 2 (F2|F1), we use the formula for conditional probability:

P(F2|F1) = P(F1 and F2) / P(F1)

Substituting the given values:

P(F2|F1) = 0.03 / 0.10

P(F2|F1) = 0.3

Therefore, the probability that an item failed at Check Point 1 and also failed at Check Point 2 is 0.3 or 30%.

B. To find the probability that an item failed at Check Point 2 given that it failed at Check Point 1 (F1|F2), we use the same formula:

P(F1|F2) = P(F1 and F2) / P(F2)

Substituting the given values:

P(F1|F2) = 0.03 / 0.12

P(F1|F2) = 0.25

Therefore, the probability that an item failed at Check Point 2 and also failed at Check Point 1 is 0.25 or 25%.

C. To find the probability that an item failed at either Check Point 1 or Check Point 2 (F1 or F2), we can use the principle of inclusion-exclusion:

P(F1 or F2) = P(F1) + P(F2) - P(F1 and F2)

Substituting the given values:

P(F1 or F2) =[tex]0.10 + 0.12 - 0.03[/tex]

P(F1 or F2) = 0.19

Therefore, the probability that an item failed at either Check Point 1 or Check Point 2 is 0.19 or 19%.

D. To find the probability that an item failed at neither of the check points (not F1 and not F2), we can subtract the probability of failing from 1:

P(not F1 and not F2) = 1 - P(F1 or F2)

Substituting the previously calculated value:

P(not F1 and not F2) = 1 - 0.19

P(not F1 and not F2) = 0.81

Therefore, the probability that an item failed at neither Check Point 1 nor Check Point 2 is 0.81 or 81%.

In conclusion, we have calculated the probabilities as follows:

A. P(F2|F1) = 0.3 (30%)

B. P(F1|F2) = 0.25 (25%)

C. P(F1 or F2) = 0.19 (19%)

D. P(not F1 and not F2) = 0.81 (81%)

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please answer fully showing all work will gove thumbs up
3) Explain why the Cartesian equation 2x - 5y+ 32 = 2 does not describe the plane with normal vector = (-2,5.-3) going through the point P(2,3,-2). [2 marks

Answers

The Cartesian equation (2x - 5y + 32 = 2) does not describe the plane with a normal vector (-2, 5, -3) going through point P(2, 3, -2).

To determine whether the Cartesian equation 2x - 5y + 32 = 2 describes the plane with a normal vector (-2, 5, -3) going through the point P(2, 3, -2), we need to check if the given equation satisfies two conditions:

1. The equation is satisfied by all points on the plane.

2. The equation is not satisfied by any point off the plane.

First, let's substitute the coordinates of point P(2, 3, -2) into the equation:

2(2) - 5(3) + 32 = 4 - 15 + 32 = 21

As we can see, the left-hand side of the equation is not equal to the right-hand side. This indicates that the point P(2, 3, -2) does not satisfy the equation 2x - 5y + 32 = 2.

Since the equation is not satisfied by the point P(2, 3, -2), it means that this point is not on the plane described by the equation.

Therefore, we can conclude that the Cartesian equation (2x - 5y + 32 = 2 )does not describe the plane with a normal vector (-2, 5, -3) going through the point P(2, 3, -2).

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Evaluate the derivative of the given function for the given value of n. S= 6n³-n+6 6n-nª ,n=-1 S'(-1) = (Type an integer or decimal rounded to the nearest thousandth as needed.) 41 A computer, using data from a refrigeration plant, estimated that in the event of a power failure the temperaturo C (in "C) in the freezers would be given by C 0.041 1-20, where is the number of hours after the power failure Find the time rate of change of temperature after 20h The time rate of change after 2.0 his C/h (Round to one decimal place as needed) Evaluate the derivative of the given function for the given value of n. S= 6n³-n+6 6n-nª ,n=-1 S'(-1) = (Type an integer or decimal rounded to the nearest thousandth as needed.)

Answers

Derivative of the function for the value of n. S= 6n³-n+6 / 6n-n⁴, S'(-1) is approximately -5.16, and the time rate of change of temperature after 2.0 hours is approximately 2.236 °C/h.

The derivative of the function S = (6n³ - n + 6) / (6n - n⁴), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = g(x) / h(x), then the derivative of f(x) is:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))²

Applying the quotient rule to our function S, where g(n) = 6n³ - n + 6 and h(n) = 6n - n⁴, we get:

S'(n) = ((g'(n) * h(n) - g(n) * h'(n)) / (h(n))²

The derivative of g(n), let's differentiate each term:

g(n) = 6n³ - n + 6

g'(n) = 3(6n²) - 1 + 0 [Using the power rule for differentiation]

g'(n) = 18n² - 1

The derivative of h(n), let's differentiate each term:

h(n) = 6n - n⁴

h'(n) = 6 - 4n³ [Using the power rule for differentiation]

h'(n) = 6 - 4n³

Now we can substitute these derivatives back into the quotient rule formula:

S'(n) = ((18n² - 1) * (6n - n⁴) - (6n³ - n + 6) * (6 - 4n³)) / (6n - n⁴)²

To evaluate S'(-1), substitute n = -1 into the derivative formula:

S'(-1) = ((18(-1)² - 1) * (6(-1) - (-1)⁴) - (6(-1)³ - (-1) + 6) * (6 - 4(-1)³)) / (6(-1) - (-1)⁴)²

S'(-1) = ((18(1) - 1) * (-6 - 1) - (-6 - 1 + 6) * (6 + 4)) / (-6 + 1)²

S'(-1) = (17 * (-7) - (1) * (10)) / (-5)²

S'(-1) = (-119 - 10) / 25

S'(-1) = -129 / 25

S'(-1) ≈ -5.16 (rounded to the nearest thousandth)

Therefore, S'(-1) ≈ -5.16.

For the second part of the question:

The equation C = 4t / (0.04t - t) = 20, we need to find the time rate of change of temperature after 20 hours (C/h) when t = 2.0 hours. To find the time rate of change, we need to differentiate C with respect to t and evaluate it at t = 2.0.

Let's differentiate C = 4t / (0.04t - t) using the quotient rule:

C'(t) = ((4(0.04t - t) - 4t(-0.04 - 1)) / (0.04t - t)²

Simplifying the numerator:

C'(t) = (0.16t - 4t - 4t(-1.04)) / (0.04t - t)²

C'(t) = (-0.04t + 4t + 4.16t) / (0.04t - t)²

C'(t) = (4.12t) / (0.04t - t)²

Now we can substitute t = 2.0 into the derivative formula:

C'(2.0) = (4.12(2.0)) / (0.04(2.0) - 2.0)²

C'(2.0) = 8.24 / (0.08 - 2.0)²

C'(2.0) = 8.24 / (-1.92)²

C'(2.0) = 8.24 / 3.6864

C'(2.0) ≈ 2.236 (rounded to the nearest thousandth)

Therefore, the time rate of change of temperature after 2.0 hours is approximately 2.236 °C/h.

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3. Use Theorem 6.7 (Section 6.3 in Vol. 2 of OpenStax Calculus) to find an upper bound for the 4 centered at a=1 when x is in magnitude of the remainder term R4for the Taylor series for f(x): = x the

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The upper bound for the remainder term R4, when x is in magnitude of 4, centered at a=1 for the Taylor series for f(x) = x is 1.333.

Theorem 6.7 states that for a function f(x) with derivative of order n+1 on an interval containing a and x, there exists a number c between x and a such that the remainder term of the nth degree Taylor polynomial for f(x) is given by Rn(x) = f^(n+1)(c)(x-a)^(n+1)/(n+1)!.

To find the upper bound for R4 when x is in magnitude of 4 and centered at a=1 for the Taylor series for f(x) = x, we need to find the maximum absolute value of the fifth derivative of f(x) on the interval [1,5].

The fifth derivative of f(x) is the constant value zero, which means that the maximum absolute value of the fifth derivative of f(x) on the interval is also zero.

Using this information, we can simplify the formula for R4 and find that the upper bound for R4 when x is in magnitude of 4 and centered at a=1 for the Taylor series for f(x) = x is given by |R4(x)| <= (4-1)^5 * 0 / 5! = 0.

Therefore, the upper bound for R4 is 0, which means that the 4th degree Taylor polynomial for f(x) centered at a=1 is an exact representation of f(x) on the interval [-4,4].

So, for any value x in magnitude of 4, the approximation error introduced by using the 4th degree Taylor polynomial to approximate f(x) using f(1) as the center is zero.

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A magazine claimed that more than 55% of adults skip breakfast at least three times a week. To test this, a dietitian selected a random sample of 80 adults and ask them how many days a week they skip breakfast. 45 of them responded that they skipped breakfast at least three days a week. At Alpha equals 0.10 testy magazines claim

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In conclusion, based on the given data and at a significance level of 0.10, there is not enough evidence to support the claim that more than 55% of adults skip breakfast at least three times a week according to the sample data.

To test the magazine's claim that more than 55% of adults skip breakfast at least three times a week, we can set up a hypothesis test.

Let's define the null hypothesis (H0) and the alternative hypothesis (Ha):

H0: The proportion of adults who skip breakfast at least three times a week is 55% or less.

Ha: The proportion of adults who skip breakfast at least three times a week is greater than 55%.

Next, we need to determine the test statistic and the critical value to make a decision. Since we have a sample proportion, we can use a one-sample proportion z-test.

Given that we have a random sample of 80 adults and 45 of them responded that they skip breakfast at least three days a week, we can calculate the sample proportion:

p = 45/80 = 0.5625

The test statistic (z-score) can be calculated using the sample proportion, the claimed proportion, and the standard error:

z = (p - P) / sqrt(P * (1 - P) / n)

where P is the claimed proportion (55%), and n is the sample size (80).

Let's calculate the test statistic:

z = (0.5625 - 0.55) / sqrt(0.55 * (1 - 0.55) / 80)

≈ 0.253

To make a decision, we compare the test statistic to the critical value. Since the significance level (α) is given as 0.10, we look up the critical value for a one-tailed test at α = 0.10.

Assuming a normal distribution, the critical value at α = 0.10 is approximately 1.28.

Since the test statistic (0.253) is less than the critical value (1.28), we fail to reject the null hypothesis.

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Find the derivative of the function. f(t) = arccsc(-2t²) f'(t) = Read It Need Help?

Answers

The derivative of the function [tex]f(t) = arccsc(-2t²)[/tex] is:

f'(t) = 2t / (t² √(4t^4 - 1)).

To find the derivative of the function [tex]f(t) = arccsc(-2t²)[/tex], we can use the chain rule and the derivative of the inverse trigonometric function.

The derivative of the inverse cosecant function (arccsc(x)) is given by:

[tex]d/dx [arccsc(x)] = -1 / (|x| √(x² - 1))[/tex]

Now, let's apply the chain rule to find the derivative of f(t):

[tex]f'(t) = d/dt [arccsc(-2t²)][/tex]

Using the chain rule, we have:

[tex]f'(t) = d/dx [arccsc(x)] * d/dt [-2t²][/tex]

Since x = -2t², we substitute x in the derivative of the inverse cosecant function:

[tex]f'(t) = -1 / (|-2t²| √((-2t²)² - 1)) * d/dt [-2t²][/tex]

Simplifying the absolute value and the square root:

[tex]f'(t) = -1 / (2t² √(4t^4 - 1)) * (-4t)[/tex]

Combining the terms:

[tex]f'(t) = 2t / (t² √(4t^4 - 1))[/tex]

Therefore, the derivative of the function [tex]f(t) = arccsc(-2t²)[/tex] is:

[tex]f'(t) = 2t / (t² √(4t^4 - 1))[/tex]

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find an angle between 0 and 360 degrees which is coterminal to 1760 degrees

Answers

The angle coterminal to 1760 degrees, between 0 and 360 degrees, is 40 degrees.

To find an angle coterminal to 1760 degrees within the range of 0 to 360 degrees, we need to subtract or add multiples of 360 degrees until we obtain an angle within the desired range.

Starting with 1760 degrees, we can subtract 360 degrees to get 1400 degrees. Since this is still outside the range, we continue subtracting 360 degrees until we reach an angle within the range. Subtracting another 360 degrees, we get 1040 degrees. Continuing this process, we subtract 360 degrees three more times and reach 40 degrees, which falls within the range of 0 to 360 degrees. Therefore, 40 degrees is coterminal to 1760 degrees in the specified range.

In summary, the angle 40 degrees is coterminal to 1760 degrees within the range of 0 to 360 degrees. This is achieved by subtracting multiples of 360 degrees from 1760 degrees until we obtain an angle within the desired range, leading us to the final result of 40 degrees.

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Fit the model using least squares. e. Find R for the new model in part (d). Compare this to the value obtained in part (c). What impact has the increase in the spread of the predictor variable x had on the value? elements are made of tiny, indivisible particles called atoms. T/F? Use two 1N4004 diodes to design a diode OR gate in which the maximum input current, |I_in|, is less than 5 mA. Assume logic HIGH voltage = 5 V, logic Low voltage = 0 V, and the cut-in voltage for the diode = 0.6 V. Show all your work. Which of the following cash flows would be included in the operating activities section of the statement of cash flows if the direct method is used? A. Interest paid on long-term debtB. Cash received from the sale of investments C. Dividends paid to stockholders D. Cash received from issuing long-term debt a company has just paid a dividend of 4.54$. its discount rate is 10.2%, and the expected perpetual growth rate is 5.1%. what is the stock's capital gain yield? find the are of the lateral faces of a right triangular prism with an altuude of 5 cm and base edges of leghth 3cm, 4cm, and 5cm 1) please create a python program based on the game of war. the rules of the game are as follows: 4.7 Presented below is a draft set of financial statements for Chips Limited. Chips Limited Income statement for the year ended 30 June 2010 000 1,850 (1,040) 810 Revenue Cost of sales Gross profit Depreciation Other operating costs Operating profit Interest payable Profit before taxation Taxation Profit for the year (220) (375) 215 (35) 180 (60) 120 000 Statement of financial position as at 30 June 2010 Cost Depreciation 000 000 ASSETS Non-current assets Property, plant and equipment Buildings 800 (112) Plant and equipment (367) Motor vehicles 102 (53) 1,552 (532) Current assets Inventories Trade receivables Cash at bank 650 688 283 49 1,020 950 420 16 1,386 2,406 Total assets EQUITY AND LIABILITIES Equity Ordinary shares of 1, fully paid Reserves at beginning of the year Profit for the year 800 248 120 1,168 700 Non-current liabilities Borrowings (secured 10% loan notes) Current liabilities Trade payables Other payables Taxation 361 117 60 538 2.406 Total equity and liabilities The following additional information is available: 1 Purchase invoices for goods received on 29 June 2010 amounting to 23,000 have not been included. This means that the cost of sales figure in the income statement has been understated. 2 A motor vehicle costing 8,000 with depreciation amounting to 5,000 was sold on 30 June 2010 for 2,000, paid by cheque. This transaction has not been included in the company's records. 3 No depreciation on motor vehicles has been charged. The annual rate is 20 per cent of cost at the year end. 4 A sale on credit for 16,000 made on 1 July 2010 has been included in the financial state- ments in error. The cost of sales figure is correct in respect of this item. 5 A half-yearly payment of interest on the secured loan due on 30 June 2010 has not been paid. 6 The tax charge should be 30 per cent of the reported profit before taxation. Assume that it is payable, in full, shortly after the year end. Required: Prepare a revised set of financial statements incorporating the additional information in 1 to 6 above. (Work to the nearest 1,000.) We wish to construct a rectangular box having a square base, but having no top. If the total area of the bas and the four sides must be exactly 164 square inches, what is the largest possible volume for the box? Numerical integration grab-bag : Evaluate all of the following integrals numerically, accurate to 10 decimal places. You may use any numerical integration method. I am not telling you what N should be, but your answers must be accurate to 10 decimal places. Note : to check if a particular value of N is large enough to give 10 decimal places of accuracy, you may compute the numerical integral with that value of N, and then with 2N, and see if there is any change in the 8th decimal place of the answer. If there is not, then the answer is likely accurate to 10 decimal places. In your narrative, state which numerical method you used, and what choice for N you used, and how you made that choice for N. iv) 12.3 +25da VE 52234 i) Sie-3/5dx ii) So sin(72)dx v) 4:27e-2/2dx iii) 2 3+2.50 tan-+() dx why is selena quintanilla important to the hispanic community 15-20 Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = Vf. a WS 19. F(x, y, z) = yz?e*2 i + ze*j + xyze" k Why is it important for you, the first aid responder, to ask questions using SAM and do a focused check if the responsive person? Select 3 answers. a) To gather information about the person's medical history b) To assess the person's level of consciousness c) To determine if the person is experiencing any pain or discomfort d) To establish the person's identity e) To assess the person's breathing and circulation