Max, Maria, and Armen were a team in a relay race. Max ran his part in 17. 3 seconds. Maria was

0. 7 seconds slower than Max. Armen was 1. 5 seconds slower than Maria. What was the total time

for the team?

Answers

Answer 1

The total time for the team in the relay race is 49 seconds.

To find the total time for the team in the relay race, we need to add the individual times of Max, Maria, and Armen.

Given that Max ran his part in 17.3 seconds, Maria was 0.7 seconds slower than Max, and Armen was 1.5 seconds slower than Maria, we can calculate their individual times:

Maria's time = Max's time - 0.7 = 17.3 - 0.7 = 16.6 seconds

Armen's time = Maria's time - 1.5 = 16.6 - 1.5 = 15.1 seconds

Now, we can find the total time for the team by adding their individual times:

Total time = Max's time + Maria's time + Armen's time

Total time = 17.3 + 16.6 + 15.1

Total time = 49 seconds

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QUESTION 17.1 POINT Find the following antiderivative: (281-x² + 3) de Do not include the constant "+" in your answer. For example, if you found the antiderivative was 2x + C you would enter 2x Provi

Answers

The antiderivative of (281 - x² + 3) is (284x - (1/3) * x³) + C, where C is the constant of integration.

How to calculate the value

Let's integrate each term:

∫(281 - x² + 3) dx

= ∫281 dx - ∫x² dx + ∫3 dx

The integral of a constant is simply the constant multiplied by x:

= 281x - ∫x² dx + 3x

= 281x - (1/3) * x^(2+1) + 3x

Simplifying the exponent:

= 281x - (1/3) * x³ + 3x

Now we can combine the terms:

= 281x + 3x - (1/3) * x³

= (284x - (1/3) * x^3) + C

So, the antiderivative of (281 - x² + 3) is (284x - (1/3) * x³) + C, where C is the constant of integration.

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Find lim f(x) and lim f(x) for the given function and value of c. X→C* X-C™ f(x) = (x+15)- |x+11/ x+11 c=-11 lim (x+15)- x-11+ |x + 111 X+11 = [ (Simplify your answer.) lim (x+15)- +11=(Simplify y

Answers

The limit of f(x) as x approaches -11 is undefined. The limit of f(x) as x approaches -11 from the right does not exist.

In the given function, f(x) = (x+15) - |x+11| / (x+11). When evaluating the limit as x approaches -11, we need to consider both the left and right limits.

For the left limit, as x approaches -11 from the left, the expression inside the absolute value becomes x+11 = (-11+11) = 0. Therefore, the denominator becomes 0, and the function is undefined for x=-11 from the left.

For the right limit, as x approaches -11 from the right, the expression inside the absolute value becomes x+11 = (-11+11) = 0. The numerator becomes (x+15) - |0| = (x+15). The denominator remains 0. Therefore, the function is also undefined for x=-11 from the right.

In summary, the limit of f(x) as x approaches -11 is undefined, and the limit from both the left and right sides does not exist due to the denominator being 0 in both cases.

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I need these Q A And B please do jot do just 1
thanks
7 Find dy dx for each of the following. x3 1 X-5 क b) 4x+3 2

Answers

7 Find dy dx for each of the following. x3 1 X-5 क b) 4x+3 2. By using the quotient rule and power rule the correct answer is (dy/dx)(4x+3/2) = 4.

Given, x^3 -1/x-5
Using the quotient rule of differentiation, we have
(dy/dx)[(x^3 -1)/(x-5)] = [(x-5)d/dx(x^3 -1) - (x^3 -1)d/dx(x-5)] / (x-5)^2
Let's find the values of d/dx(x^3 -1) and d/dx(x-5)
d/dx(x^3 -1) = 3x^2
d/dx(x-5) = 1
Now, substituting the values of d/dx(x^3 -1) and d/dx(x-5), we get
(dy/dx)[(x^3 -1)/(x-5)] = [(x-5)×3x^2 - (x^3 -1)×1] / (x-5)^2
(dy/dx)[(x^3 -1)/(x-5)] = [(3x^3 -5x^2 -1) / (x-5)^2]...ans
Let's find dy/dx for 4x+3/2
Using the power rule of differentiation, we have
(dy/dx)(4x+3/2) = 4(d/dx)(x) + d/dx(3/2)
(dy/dx)(4x+3/2) = 4 + 0
(dy/dx)(4x+3/2) = 4 ...ans

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Let T: R3 + R2 be the map TT (x, y, z) + (x2 + yz, ecyz) and w be the 2-form w = uvụ du 1 dv = Then calculate and simplify the following TW T*w Next, use this to calculate and simplify the following d(7*w) Do not use the fact that d(*W) = ** (dw). =

Answers

To calculate TW, substitute the coordinates (x, y, z) into T(x, y, z) = (x²+ yz, e^cyz). For Tˣw, substitute the coordinates (u, v) into Tˣw = u(x^2 + yz)dv. To calculate d(7ˣw), differentiate 7ˣw using exterior differentiation: d(7ˣw) = 7(du∧v + udv∧dv).

What is the calculation process for TW, Tˣw, and d(7ˣw) in the given scenario?

The map T: R³ → R²  is defined as T(x, y, z) = (x²   + yz, e^cyz), and the 2-form w is given as w = uvdv.

To calculate TW, we substitute the coordinates (x, y, z) into the map T and obtain T(x, y, z) = (x²   + yz, e^cyz).

Next, we calculate T³w by substituting the coordinates (u, v) into the 2-form w. Since w = uvdv, we have Tˣw = u(x²   + yz)dv.

To calculate d(7ˣw), we differentiate the 2-form 7ˣw. Since w = uvdv, we have d(7ˣw) = d(7uvdv). Using the properties of exterior differentiation, we obtain d(7ˣw) = 7d(uv)∧dv = 7(du∧v + udv∧dv).

It's important to note that we are not using the fact that d(ˣw) = ˣˣ(dw) in this calculation.

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Which statement is true

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The correct statement is:

D) One of its factors is x + 1.

To find the roots, we set the polynomial equal to zero:

x⁴ + x³ -3x² -5x- 2= 0

However, based on the given options, we can check which option satisfies the given conditions. Let's evaluate each option:

A) Two of its factors are x + 1

If two factors are x + 1, it means that (x + 1) is a factor repeated twice. This would imply that the polynomial has a double root at x = -1.

We can verify this by substituting x = -1 into the polynomial:

(-1)⁴ + (-1)³ - 3(-1)² - 5(-1) - 2 = 1 - 1 - 3 + 5 - 2 = 0

The polynomial indeed evaluates to zero at x = -1, so this option is plausible.

B) All four of its factors are x + 1

If all four factors are x + 1, it means that (x + 1) is a factor repeated four times. However, we have already established that the polynomial has a double root at x = -1. Therefore, this option is not correct.

C) Three of its factors are x + 1

Similar to option B, if three factors are x + 1, it implies that (x + 1) is a factor repeated three times. However, we know that the polynomial has a double root at x = -1, so this option is also incorrect.

D) One of its factors is x + 1

If one factor is x + 1, it means that (x + 1) is a distinct root or zero of the polynomial. We have already established that x = -1 is a root, so this option is plausible.

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Use the information given about the angle 0, 0 50 2r., to find the exact value of each trigonometric function.
sec 0 = 9 sino> 0

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To find the exact values of each trigonometric function, we need to solve for the angle 0 using the given information. From the equation sec 0 = 9 sin 0, we can rewrite it in terms of cosine and sine:

sec 0 = 1/cos 0 = 9 sin 0

To simplify the equation, we can square both sides:

(1/cos 0)^2 = (9 sin 0)^2

1/cos^2 0 = 81 sin^2 0

Using the Pythagorean identity sin^2 0 + cos^2 0 = 1, we can substitute 1 - sin^2 0 for cos^2 0:

1/(1 - sin^2 0) = 81 sin^2 0

Now, let's solve for sin^2 0:

81 sin^4 0 - 81 sin^2 0 + 1 = 0

This is a quadratic equation in sin^2 0. Solving it, we find:

sin^2 0 = (81 ± √(6560))/162

Since sin^2 0 cannot be negative, we discard the negative square root. Therefore:

sin^2 0 = (81 + √(6560))/162

Now, we can find sin 0 by taking the square root:

sin 0 = √((81 + √(6560))/162)

With the value of sin 0, we can find the exact values of other trigonometric functions using the identities:

cos 0 = √(1 - sin^2 0)

tan 0 = sin 0 / cos 0

cosec 0 = 1 / sin 0

cot 0 = 1 / tan 0

Substituting the value of sin 0 obtained, we can calculate the exact values for each trigonometric function.

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please using product rule
2. Find the derivative of each of the following. Simplify each answer to ensure no negative exponents remain. a) y = (2√x - 3)(6 - 5x¹) b) y = (-/-) (¹² + ⁹) 3. Find the equation of the tangent

Answers

a) To find the derivative of y = (2√x - 3)(6 - 5x), we can use the product rule. Applying the product rule, we have:

y' = (2)(6 - 5x) + (2√x - 3)(-5)

Simplifying further, we get:

y' = 12 - 10x - 10√x + 15

Combining like terms, the simplified derivative is:

y' = -10x - 10√x + 27

b) To find the derivative of y = (-/-) (12 + 9)³, we can apply the power rule. The power rule states that for a function of the form f(x) = ax^n, the derivative is given by f'(x) = nax^(n-1).

Applying the power rule, we have:

y' = (-/-) (3)(12 + 9)^(3-1)

Simplifying further, we get:

y' = (-/-) (3)(21)^2

The derivative simplifies to:

y' = (-/-) 1323

Therefore, the derivative of y = (-/-) (12 + 9)³ is y' = (-/-) 1323.

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how
is this solved?
Find the Taylor polynomial of degree n = 4 for x near the point a for the function sin(3x).

Answers

This is the Taylor polynomial of degree n = 4 for x near the point a for the function sin(3x). To find the Taylor polynomial of degree n = 4 for x near the point a for the function sin(3x), we need to compute the function's derivatives up to the fourth derivative at x = a.

The Taylor polynomial of degree n for a function f(x) near the point a is given by:

P(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + (f'''(a)/3!)(x - a)^3 + ... + (f^n(a)/n!)(x - a)^n,

where f'(a), f''(a), f'''(a), ..., f^n(a) represent the first, second, third, ..., nth derivatives of f(x) evaluated at x = a. In this case, the function is f(x) = sin(3x), so we need to compute the derivatives up to the fourth derivative:

f(x) = sin(3x),

f'(x) = 3cos(3x),

f''(x) = -9sin(3x),

f'''(x) = -27cos(3x),

f^4(x) = 81sin(3x).

Now we can evaluate these derivatives at x = a to obtain the coefficients for the Taylor polynomial:

f(a) = sin(3a),

f'(a) = 3cos(3a),

f''(a) = -9sin(3a),

f'''(a) = -27cos(3a),

f^4(a) = 81sin(3a).

Substituting these coefficients into the formula for the Taylor polynomial, we get:

P(x) = sin(3a) + 3cos(3a)(x - a) - (9sin(3a)/2!)(x - a)^2 - (27cos(3a)/3!)(x - a)^3 + (81sin(3a)/4!)(x - a)^4.  

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let f(x, y, z) = y² i (2xy e²) j e²y k be a vector field. a) determine whether or not f is a conservative vector field

Answers

The vector field f(x, y, z) is not a conservative vector field.

A vector field is said to be conservative if it can be expressed as the gradient of a scalar function called a potential function. In other words, if f = ∇φ, where φ is a scalar function, then the vector field f is conservative.

To determine whether the given vector field f(x, y, z) = y²i + (2xye²)j + e²yk is conservative, we need to check if its curl is zero. If the curl of a vector field is zero, then the vector field is conservative.

Taking the curl of f, we have:

curl(f) = (∂f₃/∂y - ∂f₂/∂z)i + (∂f₁/∂z - ∂f₃/∂x)j + (∂f₂/∂x - ∂f₁/∂y)k

Substituting the components of f, we get:

curl(f) = (0 - 2xe²)i + (0 - 0)j + (2xe² - y²)k

Since the curl of f is not zero (it has non-zero components), we conclude that the vector field f is not conservative.

Therefore, the given vector field f(x, y, z) = y²i + (2xye²)j + e²yk is not a conservative vector field.

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ſ 16 sin’x cos²x dx the solution is 2x - 4 sin x cosx + 2 sin x cos x +C 1 x - 2 sin x cos x + 4 sin x cos x + C 2 1 1 5 sin x + sin x + c 14 3

Answers

The solution to the integral ∫16sin(x)cos²(x) dx is 2x - 4sin(x)cos(x) + 2sin(x)cos(x) + C, where C represents the constant of integration. This can be simplified to 2x - 2sin(x)cos(x) + C.

To obtain the solution, we can use the trigonometric identity cos²(x) = (1/2)(1 + cos(2x)), which allows us to rewrite the integrand as 16sin(x)(1/2)(1 + cos(2x)). We then expand and integrate each term separately. The integral of sin(x) dx is -cos(x) + C, and the integral of cos(2x) dx is (1/2)sin(2x) + C. By substituting these results back into the expression and simplifying, we arrive at the final solution.

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[-/1 Points! DETAILS WAI Calculate the consumers surplus at the indicated unit price p for the demand equation. HINT (See Example 1.] (Round your answer to the nearest cent.) p = 80 - 9; p = 20 $ Need

Answers

We must determine the region between the demand curve and the price line in order to compute the consumer surplus at the unit.

price p for the demand equation p = 80 - 9 with p = 20.

Rewriting the demand equation as  - 9p, where q stands for the quantity demanded.

We can replace the supplied price, p = 20, into the demand equation to determine the corresponding quantity demanded:

[tex]q = 80 - 9(20) = 80 - 180 = -100.[/tex]

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8. (4 pts) Let m= (1, 2, 3) and n=(5. 3.-2). Find the vector projection of monton, that is, find proj, m. You do not need to simplify (radicals in denominators are okay).

Answers

The vector projection of vector m onto vector n can be found by taking the dot product of m and n, dividing it by the magnitude of n squared, and then multiplying the result by vector n.

To find the vector projection of m onto n, we first need to calculate the dot product of m and n. The dot product of two vectors is obtained by multiplying their corresponding components and summing them up. In this case, the dot product of m and n is calculated as (1 * 5) + (2 * 3) + (3 * -2) = 5 + 6 - 6 = 5.

Next, we need to find the magnitude of n squared. The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. In this case, the magnitude of n squared is calculated as [tex](5^2) + (3^2) + (-2^2) = 25 + 9 + 4 = 38[/tex].

Finally, we can calculate the vector projection by dividing the dot product of m and n by the magnitude of n squared and then multiplying the result by n. So, the vector projection of m onto n is (5 / 38) * (5, 3, -2) = (25/38, 15/38, -10/38).

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Find the volume of the solid obtained by rotating the region bounded by Y=3x +2 y=x2+2 x=0 Rotating X=2 Washer method OR Disc Method

Answers

1) The intersection points are x = 0 and x = 3. These will be our limits of integration.

2)  R = distance from x-axis to outer curve[tex]= 3x + 2 - 2 = 3x[/tex]

    r = distance from x-axis to inner curve =[tex]x^2 + 2 - 2 = x^2[/tex]

3) V = π ∫[tex](0 to 3) (9x^2 - x^4) dx[/tex]

4) V = π [27 - 81/5]

5) V = (54/5)π

How to find the volume?

To find the volume of the solid obtained by rotating the region bounded by the curves [tex]y = 3x + 2, y = x^2 + 2[/tex], and x = 0 using the washer method (or disc method) about the line x = 2, we can follow these steps:

1. Determine the limits of integration:

  The region is bounded by[tex]y = 3x + 2[/tex] and [tex]y = x^2 + 2[/tex]. To find the limits of integration for x, we need to determine the x-values at which the two curves intersect.

 

  Setting the two equations equal to each other:

   [tex]3x + 2 = x^2 + 2[/tex]

 

  Rearranging and simplifying:

  [tex]x^2 - 3x = 0[/tex]

 

  Factoring:

  x(x - 3) = 0

 

Therefore, the intersection points are x = 0 and x = 3. These will be our limits of integration.

2. Determine the radius of each washer:

  The washer method involves finding the difference in areas of two circles: the outer circle and the inner circle.

  The outer radius (R) is the distance from the axis of rotation (x = 2) to the outer curve [tex](y = 3x + 2).[/tex]

  The inner radius (r) is the distance from the axis of rotation (x = 2) to the inner curve[tex](y = x^2 + 2)[/tex]

  The formula for the outer and inner radii is:

  R = distance from x-axis to outer curve[tex]= 3x + 2 - 2 = 3x[/tex]

  r = distance from x-axis to inner curve =[tex]x^2 + 2 - 2 = x^2[/tex]

3. Set up the integral for the volume using the washer method:

  The volume of each washer is given by: π[tex][(R^2) - (r^2)]dx[/tex]

 

The volume of the solid can be calculated by integrating the volumes of all the washers from x = 0 to x = 3:

  V = ∫(0 to 3) π[tex][(3x)^2 - (x^2)^2]dx[/tex]

  Simplifying:

  V = π ∫[tex](0 to 3) (9x^2 - x^4) dx[/tex]

4. Evaluate the integral:

  Integrating the expression, we get:

  V = π [tex][3x^3/3 - x^5/5][/tex] evaluated from 0 to 3

  V = π[tex][(3(3)^3/3 - (3)^5/5) - (3(0)^3/3 - (0)^5/5)][/tex]

  V = π [27 - 81/5]

5. Finalize the volume:

  Simplifying the expression, we have:

  V = π [(135/5) - (81/5)]

  V = π (54/5)

  V = (54/5)π

Therefore, the volume of the solid obtained by rotating the region bounded by [tex]y = 3x + 2, y = x^2 + 2[/tex], and x = 0 about the line x = 2 using the washer method is (54/5)π cubic units.

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Use the four-step process to find f'(x) and then find f'(1), f'(2), and f'(3). 7 f(x) = 6 + х f'(x) = x) = C

Answers

Answer:

using four step process we found that f'(1) = 1, f'(2) = 1, and f'(3) = 1.

Step-by-step explanation:

To find f'(x), the derivative of f(x), we can apply the four-step process:

Identifying the function f(x).

f(x) = 6 + x

Apply the power rule of differentiation.

For any constant C, the derivative of C with respect to x is 0.

The derivative of x with respect to x is 1.

Combine the derivatives obtained in Step 2.

Since the derivative of a constant is 0, we only need to consider the derivative of x.

f'(x) = 0 + 1

      = 1

Step 4: Evaluate f'(x) at the given values of x.

  f'(1) = 1

  f'(2) = 1

  f'(3) = 1

Therefore, f'(1) = 1, f'(2) = 1, and f'(3) = 1.

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Convert the equation to polar form. (use variables and needed) MY OTH ron 1 sin 0 seco 3 X x

Answers

The equation is in polar form, where r is the distance from the origin and θ is the angle. The equation is:

-2r cos(θ) = 1

To convert the equation to polar form, we need to express the variables x and y in terms of polar coordinates. In polar coordinates, a point is represented by its distance from the origin (r) and the angle it makes with the positive x-axis (θ).

Here,

x = r cos(θ)

y = r sin(θ)

We have the equation:

x - 1 = sin(0) + 3x

Substituting the expressions for x and y in terms of polar coordinates, we get:

r cos(θ) - 1 = sin(0) + 3(r cos(θ))

Let's simplify this equation:

r cos(θ) - 1 = 0 + 3r cos(θ)

Rearranging the terms:

r cos(θ) - 3r cos(θ) = 1

Combining like terms:

-2r cos(θ) = 1

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during a single day at radio station wmzh, the probability that a particular song is played is 50%. what is the probability that this song will be played on 2 days out of 4 days? round your answer to

Answers


The probability of a song being played on a single day is 0.5. We need to find the probability of the song being played on 2 days out of 4 days. This can be solved using the binomial probability formula, which is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful events, p is the probability of success, and (n choose k) is the binomial coefficient. Substituting the values, we get P(X=2) = (4 choose 2) * 0.5^2 * 0.5^2 = 0.375. Therefore, the probability that this song will be played on 2 days out of 4 days is 0.375.

The problem can be solved using the binomial probability formula because we are interested in finding the probability of a particular event (the song being played) occurring a specific number of times (2 out of 4 days) in a fixed number of trials (4 days).

We use the binomial probability formula P(X=k) = (n choose k) * p^k * (1-p)^(n-k) to calculate the probability of k successful events occurring in n trials with a probability of success p.

In this case, n=4, k=2, p=0.5. Therefore, P(X=2) = (4 choose 2) * 0.5^2 * 0.5^2 = 0.375.

The probability that a particular song will be played on 2 days out of 4 days at radio station wmzh is 0.375 or 37.5%.

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II. Calculations and Applications: 1. The demand equation for a certain product is x = p + 30, where p is the unit price and x is the quantity demanded of the product. Find the elasticity of demand an

Answers

The demand is inelastic at a price of 5 and elastic at a price of 10.

To find the elasticity of demand, we need to calculate the derivative of the demand equation with respect to the unit price (p) and then evaluate it at the indicated prices. The elasticity of demand is given by the formula:

Elasticity of Demand = (dX/dP) * (P/X)

Let's calculate the elasticity at the indicated prices:

Elasticity at Price p = 5:

To find the quantity demanded (x) at this price, we substitute p = 5 into the demand equation:

x = (-5/2)(5) + 30

x = -25/2 + 30

x = -25/2 + 60/2

x = 35/2

Now, let's find the derivative of the demand equation:

dX/dP = -5/2

Now we can calculate the elasticity:

Elasticity at p = 5 = (-5/2) * (5 / (35/2))

Elasticity at p = 5 = (-5/2) * (2/7)

Elasticity at p = 5 = -5/7

Since the elasticity is less than 1, the demand is inelastic at a price of 5.

Elasticity at Price p = 10:

To find the quantity demanded (x) at this price, we substitute p = 10 into the demand equation:

x = (-5/2)(10) + 30

x = -50/2 + 30

x = -50/2 + 60/2

x = 10/2

x = 5

Now, let's find the derivative of the demand equation:

dX/dP = -5/2

Now we can calculate the elasticity:

Elasticity at p = 10 = (-5/2) * (10 / 5)

Elasticity at p = 10 = (-5/2) * 2

Elasticity at p = 10 = -5

Since the elasticity is equal to -5, which is greater than 1 (in absolute value), the demand is elastic at a price of 10.

Therefore, the demand is inelastic at a price of 5 and elastic at a price of 10.

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Incomplete question:

The demand equation for certain products is x = (-5/2)p+ 30 where p is the unit price and  x is the quantity demanded of the product. Find the elasticity of demand and determine whether the demand is elastic or inelastic at the indicated prices:

Suppose we have a loaded die that gives the outcomes 1 through 6 according to the following probability distribution. Pips Showing 1 2 3 4 5 6 Probability 0.1 0.2 0.3 0.2 ? 0.1 Find the probability of rolling a 5.

Answers

The probability of rolling a 5 is 0.1.

To find the missing probability for rolling a 5, we can use the fact that the sum of all probabilities for all possible outcomes must equal 1.

Let's calculate the missing probability:

1. Sum the probabilities of the given outcomes: 0.1 + 0.2 + 0.3 + 0.2 + 0.1 = 0.9.

2. Subtract the sum from 1 to find the missing probability: 1 - 0.9 = 0.1.

Therefore, the missing probability for rolling a 5 is 0.1 or 10%.

Here are the steps summarized:

1. Calculate the sum of the given probabilities: 0.1 + 0.2 + 0.3 + 0.2 + 0.1 = 0.9.

2. Subtract the sum from 1 to find the missing probability: 1 - 0.9 = 0.1.

This approach ensures that the probabilities for all possible outcomes in the probability distribution add up to 1, as required. In this case, the sum of all probabilities is 0.9, so the missing probability for rolling a 5 is the remaining 0.1 or 10% needed to reach a total probability of 1.

Hence, the probability of rolling a 5 is 0.1 or 10%.

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Let l be the line containing (0,0,1) that is parallel to y = 2x is the xy-plane. a. Sketch the line L 1 write its equation in parametric vector form b. Let P be the plane containing 2010, 1) that is perpen- dicules to live L. Include ? in your sketch from part a. Find the equation for P. c. Let Po be a point on line L,Pot 50 10,1). Find a L point Pot that is on L, the same distance from (0,01) as Po, and is on the other side of slave P from Po.

Answers

The values of all sub-parts have been obtained.

(a). The equation of the line in parametric vector form is vec-tor-r = (2λ, λ, 1).

(b). The equation of the plane P is 2x + y = 0.

(c). The value of point P₀ is (-2, -1, 1).

What is parametric form of equation?

Equation of this type is known as a parametric equation; it uses an independent variable known as a parameter (commonly represented by t) and dependent variables that are defined as continuous functions of the parameter and independent of other variables. When necessary, more than one parameter can be used.

(a). Evaluate the equation of the line in parametric vec-tor form:

Now the direction is along the line y = 2x in xy-plane. Also the line is passing through (0, 0, 1).

The equation of line in symmetric form is,

x/2 = y/1 = (z - 1)/0 = λ  

Then equation of the line in parametric vec-tor form is,

vec-tor-r = (2λ, λ, 1)      

(b). Evaluate the equation of the plane P:

Now direction ratios of the line L is (2, 1, 0).

So, equation of plane passing through (0, 0, 0) and perpendicular to (2, 1, 0) is,

2 (x - 0) + 1 (y - 0) + 0 (z - 1) = 0

2x + y = 0

(c). Evaluate the value of point P₀:

Let P₀ say (2, 1, 1) be a point on the line L.

Let P₀ˣ (2λ, λ, 1) be a point on the line other side of P₀ to the plane P.

Middle point (λ+1, (λ + 1)/2, 1) of P₀ˣ P₀ lies on the plane.

The middle point satisfies 2x + y = 0.

Then ,

2(λ + 1) + (λ + 1)/2 =0

4λ + 4 + λ + 1 = 0

5λ + 5 = 0

5λ = -5

λ = -1

Then substitutes (λ = -1) in P₀ˣ (2λ, λ, 1)

P₀ˣ = (-2, -1, 1).

Hence, the values of all sub-parts have been obtained.

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A child decides to sell balsa-wood gliders outside of the
Astoria column for visitors to fly from the top. She determines
that her profit is given by the function p(x)=-55-6x+0.2x^2 where
"x" is t

Answers

The profit function of the child selling balsa-wood gliders outside the Astoria column is given by p(x) = -55 - 6x + 0.2[tex]x^{2}[/tex], where "x" represents the number of gliders sold. This function represents the relationship between the profit made and the quantity of gliders sold.

The profit function p(x) = -55 - 6x + 0.2x^2 is a quadratic function with respect to the number of gliders sold, denoted by "x". The coefficients in the function represent various factors influencing the profit. The term -55 represents a fixed cost or initial investment, which will reduce the profit regardless of the number of gliders sold. The term -6x represents the variable cost associated with producing each glider. It implies that for each glider sold, the profit will decrease by $6. Finally, the term 0.2x^2 represents the revenue generated by selling gliders. As the quantity of gliders sold increases, the revenue increases quadratically.

By subtracting the costs (fixed and variable) from the revenue, we obtain the profit function. The child can determine the maximum profit by analyzing the function's vertex, which represents the optimal quantity of gliders to sell. In this case, the vertex corresponds to the maximum point on the profit function's graph, indicating the number of gliders the child should sell to maximize their profit.

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C(x) = 0.05x2 + 22x + 340, 0 < < 150. (A) Find the average cost function C(x). (B) List all the critical values of C(x). Note: If there are no critical values, enter 'NONE'. (C) Use interval notation

Answers

A) The average cost function C(x) can be obtained by dividing the total cost function by the quantity x:

C(x) = (0.05x^2 + 22x + 340) / x

Simplifying this expression, we get:

C(x) = 0.05x + 22 + 340/x

Therefore, the average cost function C(x) is given by 0.05x + 22 + 340/x.

B) To find the critical values of C(x), we need to determine the values of x where the derivative of C(x) is equal to zero or is undefined. Taking the derivative of C(x) with respect to x, we have:

C'(x) = 0.05 - 340/x^2

Setting C'(x) equal to zero and solving for x, we find:

0.05 - 340/x^2 = 0

Rearranging the equation, we have:

340/x^2 = 0.05

Simplifying further, we get:

x^2 = 340/0.05

x^2 = 6800

Taking the square root of both sides, we find two critical values:

x = ± √(6800)

Therefore, the critical values of C(x) are x = √(6800) and x = -√(6800)

C) Using interval notation, we can express the domain of x where the function C(x) is defined. Given that the range of x is from 0 to 150, we can represent this interval as (0, 150).

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problem :- - T 2 1 TIP3 P32 3 > T(f) = f' By -z , x², x3} 2 Bw = ₂ 1 n, x 2 } Find matrixe representation of line as Iransformation ? > 3

Answers

To find the matrix representation of the linear transformation T(f) = (f' - 2f, x^2, x^3) with respect to the basis {1, x, x^2, x^3}, we need to determine the transformation of each basis vector and express the results as linear combinations of the basis vectors.

The coefficients of these linear combinations form the columns of the matrix representation.

To find the matrix representation of the linear transformation T(f) = (f' - 2f, x^2, x^3) with respect to the basis {1, x, x^2, x^3}, we apply the transformation to each basis vector.

Applying the transformation T to the basis vector 1, we have T(1) = (0 - 2(1), 1^2, 1^3) = (-2, 1, 1).

Applying the transformation T to the basis vector x, we have T(x) = (d/dx(x) - 2(x), x^2, x^3) = (1 - 2x, x^2, x^3).

Applying the transformation T to the basis vector x^2, we have T(x^2) = (d/dx(x^2) - 2(x^2), (x^2)^2, (x^2)^3) = (2x - 2x^2, x^4, x^6).

Applying the transformation T to the basis vector x^3, we have T(x^3) = (d/dx(x^3) - 2(x^3), (x^3)^2, (x^3)^3) = (3x^2 - 2x^3, x^6, x^9)

Expressing each of these results as linear combinations of the basis vectors, we obtain:

(-2, 1, 1) = -2(1) + 1(x) + 1(x^2) + 0(x^3),

(1 - 2x, x^2, x^3) = 1(1) - 2(x) + 0(x^2) + 0(x^3),

(2x - 2x^2, x^4, x^6) = 0(1) + 2(x) - 2(x^2) + 0(x^3),

(3x^2 - 2x^3, x^6, x^9) = 0(1) + 0(x) + 0(x^2) + 3(x^3).

The coefficients of these linear combinations form the columns of the matrix representation of the linear transformation T with respect to the basis {1, x, x^2, x^3}. Thus, the matrix representation is:

[-2 1 0 0

1 -2 0 0

0 2 -2 3

0 0 0 0]

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2. Find all of the values of x where the following function is not continuous. For each value, state whether the discontinuity is removable or not. x2 + 2x + 1 f(x) x2 + 3x + 2 =

Answers

The function f(x) = x^2 + 2x + 1 / (x^2 + 3x + 2) is not continuous at x = -1 and x = -2. The discontinuity at x = -1 is removable because the function can be redefined at that point to make it continuous.

The discontinuity at x = -2 is non-removable because there is a vertical asymptote at that point, which cannot be removed by redefining the function. At x = -1, both the numerator and denominator of the function become zero, resulting in an indeterminate form.

By factoring both expressions, we find that f(x) can be simplified to f(x) = (x + 1) / (x + 1) = 1, which defines a single point that can replace the discontinuity. However, at x = -2, the denominator becomes zero while the numerator remains nonzero, resulting in an infinite value and a vertical asymptote. Therefore, the discontinuity at x = -2 is non-removable..

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Please answer all 3 questions, thank youuu.
2 Points Question 4 A spring has a natural length of 15 inches. A force of 10 lbs. is required to keep it stretched 5 inches beyond its natural length. Find the work done in stretching it from 20 inch

Answers

The work done in stretching the spring from 20 inches is 50 inches• lbs.

Given, A spring has a natural length of 15 inches. A force of 10 lbs. is required to keep it stretched 5 inches beyond its natural length. We have to find the work done in stretching it from 20 inches.

Here, The work done in stretching a spring can be determined by the formula, W = 1/2 kx² Where, W represents work done in stretching a spring k represents spring constant x represents distance stretched beyond natural length

Therefore, we have to first find the spring constant, k. Given force, F = 10 lbs, distance, x = 5 inches. Then k = F / x = 10 / 5 = 2The spring constant of the spring is 2.

Therefore, Work done to stretch the spring by 5 inches beyond its natural length will be, W = 1/2 kx²  W = 1/2 x 2 x 5² = 25 inches •lbs

Work done = work done to stretch the spring by 5 inches beyond its natural length + work done to stretch the spring by additional 15 inches W = 25 + 1/2 x 2 x (20 - 15)²

W = 25 + 1/2 x 2 x 5²

W = 25 + 25W = 50 inches •lbs

Hence, the work done in stretching the spring from 20 inches is 50 inches• lbs.

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Let u=5i-j+k, v=i+5k, w=-15i+3j-3k which rectors, if any, are parallel, perpendicular? Give reasons for your answer.

Answers

Only vectors v and w are perpendicular to each other.

To determine if vectors are parallel or perpendicular, we can analyze their dot products.

a) Comparing vectors u = 5i - j + k and v = i + 5k:

To check for parallelism, we'll calculate the dot product u · v:

u · v = (5i)(i) + (-j)(0) + (k)(5k)

= 5i^2 + 0 + 5k^2

= 5 + 5

= 10

Since the dot product is non-zero (10), the vectors u and v are not perpendicular.

b) Comparing vectors u = 5i - j + k and w = -15i + 3j - 3k:

To check for parallelism, we'll calculate the dot product u · w:

u · w = (5i)(-15i) + (-j)(3j) + (k)(-3k)

= -75i^2 - 3j^2 - 3k^2

= -75 - 3 - 3

= -81

Since the dot product is non-zero (-81), the vectors u and w are not perpendicular.

c) Comparing vectors v = i + 5k and w = -15i + 3j - 3k:

To check for parallelism, we'll calculate the dot product v · w:

v · w = (i)(-15i) + (5k)(3j) + (-15k)(-3k)

= -15i^2 + 15k^2

= -15 + 15

= 0

Since the dot product is zero, the vectors v and w are perpendicular.

In summary:

Vectors u and v are neither parallel nor perpendicular.

Vectors u and w are neither parallel nor perpendicular.

Vectors v and w are perpendicular.

Therefore, among the given vectors, v and w are perpendicular to each other.

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How do I do this without U-sub using trig sub
14 √ ₁ x ³ √T-x² dx J вл 0 Use Theta = arcsin to convert x bounds to theta bounds (edited)

Answers

The solution to the integral ∫(0 to 1) x³√(T - x²) dx using trigonometric substitution is [tex](3T^{(3/2)})/8[/tex].

What is trigonometry?

One of the most significant areas of mathematics, trigonometry has a wide range of applications. The study of how the sides and angles of a right-angle triangle relate to one another is essentially what the field of mathematics known as "trigonometry" is all about.

To solve the integral ∫(0 to 1) x³√(T - x³) dx using a trigonometric substitution, you can follow these steps:

Step 1: Identify the appropriate trigonometric substitution. In this case, let's use x = √T sinθ, which implies dx = √T cosθ dθ.

Step 2: Convert the given bounds of integration from x to θ. When x = 0, sinθ = 0, which gives θ = 0. When x = 1, sinθ = 1, which gives θ = π/2.

Step 3: Substitute x and dx in terms of θ in the integral:

∫(0 to π/2) (√T sinθ)³ √(T - (√T sinθ)²) (√T cosθ) dθ

= ∫(0 to π/2) [tex]T^{(3/2)}[/tex] sin³θ cos²θ dθ

Step 4: Simplify the integrand using trigonometric identities. Recall that sin²θ = 1 - cos²θ.

=[tex]T^{(3/2)}[/tex] ∫(0 to π/2) sin^3θ (1 - sin²θ) cosθ dθ

Step 5: Expand the integrand and split it into two separate integrals:

= [tex]T^{(3/2)}[/tex] ∫(0 to π/2) (sin³θ - [tex]sin^5[/tex]θ) cosθ dθ

Step 6: Integrate each term separately. The integral of sin³θ cosθ can be evaluated using a u-substitution.

Let u = sinθ, du = cosθ dθ.

= [tex]T^{(3/2)}[/tex] ∫(0 to π/2) u³ du

= [tex]T^{(3/2)} [u^{4/4}][/tex] (0 to π/2)

= [tex]T^{(3/2)} [(sinθ)^{4/4}][/tex] (0 to π/2)

= [tex]T^{(3/2)} [1/4] - T^{(3/2)} [0][/tex]

= [tex]T^{(3/2)}/4[/tex]

The integral of [tex]sin^5[/tex]θ cosθ can be evaluated using integration by parts.

Let dv = [tex]sin^5[/tex]θ cosθ dθ, u = sinθ, v = -1/6 cos²θ.

=[tex]T^{(3/2)}[/tex][-1/6 cos²θ sinθ] (0 to π/2) - [tex]T^{(3/2)}[/tex] ∫(0 to π/2) (-1/6 cos²θ) cosθ dθ

= [tex]T^{(3/2)}[/tex] [-1/6 cos²θ sinθ] (0 to π/2) + [tex]T^{(3/2)}[/tex]/6 ∫(0 to π/2) cos³θ dθ

Using the reduction formula for the integral of cos^nθ, where n is a positive integer, we have:

∫(0 to π/2) cos³θ dθ = (3/4) ∫(0 to π/2) cosθ dθ - (1/4) ∫(0 to π/2) cos³θ dθ

Rearranging the equation:

(5/4) ∫(0 to π/2) cos³θ dθ = (3/4) ∫(0 to π/2) cosθ dθ

(1/4) ∫(0 to π/2) cos³θ dθ = (3/4) ∫(0 to π/2) cosθ dθ

(1/4) ∫(0 to π/2) cos³θ dθ = (3/4) [sinθ] (0 to π/2)

= (3/4) [1 - 0]

= 3/4

Substituting back into the expression:

= [tex]T^{(3/2)}[/tex] [-1/6 cos²θ sinθ] (0 to π/2) + [tex]T^{(3/2)}/6 (3/4)[/tex]

= [tex]T^{(3/2)}[/tex] [-1/6 cos²θ sinθ] (0 to π/2) + [tex]T^({3/2)}/8[/tex]

= [tex]T^{(3/2)} [-1/6 (0) (1) - (-1/6) (1) (0)] + T^{(3/2)}/8[/tex]

=[tex]T^{(3/2)}/8[/tex]

Step 7: Combine the results from both integrals:

∫[tex](0 to 1) x^3√(T - x^2) dx = T^{(3/2)}/4 + T^{(3/2)}/8[/tex]

= [tex](3T^{(3/2)})/8[/tex]

Therefore, the solution to the integral ∫(0 to 1) x³√(T - x²) dx using trigonometric substitution is [tex](3T^{(3/2)})/8[/tex].

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-5
0
In which direction does the shape move?
A shape is translated by the vector
A
A Left
B
B Right C Up
C
D
D
Only 1 attempt allowed.

Answers

The shape moves in the direction B: Right.

When a shape is translated by a vector, the vector represents the displacement or movement of the shape.

In this case, the vector [-5, 0] indicates a movement of 5 units to the left along the x-axis and no movement along the y-axis (0 units up or down).

Since the x-axis is typically oriented from left to right, a movement of -5 units along the x-axis implies a movement to the left.

Therefore, the shape moves to the right.

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Evaluate Question 1 Not yet answered I= S. (2.42 +3. +3. 2) dx + (4.2 - y) dy Marked out of 5.00 in the c, y) plane from (0,0) to (1,4) where: P Flag question (a) C is the curvey = 4.23. I (b) C is th

Answers

The evaluated line integral in the (x, y) plane from (0,0) to (1,4) for the given options is as follows: (a) For C: y = 4x³, I = ∫[0 to 1] (2.42 + 3 + 3²) dx + ∫[0 to 4] (4.2 - y) dy, (b) For C: y = 4x, I = ∫[0 to 1] (2.42 + 3 + 3²) dx + ∫[0 to 4] (4.2 - y) dy.

(a) In option (a), we have the curve C defined as y = 4x³. We calculate the line integral I by evaluating two integrals: the first integral is with respect to x from 0 to 1, and the second integral is with respect to y from 0 to 4.

(a) For C: y = 4x³, I = ∫[0 to 1] (2.42 + 3 + 3²) dx + ∫[0 to 4] (4.2 - y) dy

= (2.42 + 3 + 3²) ∫[0 to 1] dx + ∫[0 to 4] (4.2 - 4x³) dy

= (2.42 + 3 + 3²) [x] from 0 to 1 + (4.2y - x³y) from 0 to 4

= (2.42 + 3 + 3²)(1 - 0) + (4.2(4) - 1³(4)) - (4.2(0) - 1³(0))

= (2.42 + 3 + 3²)(1) + (4.2(4) - 64)

= (2.42 + 3 + 9)(1) + (16.8 - 64)

= (14.42)(1) - 47.2

= 14.42 - 47.2

= -32.78

b) In option (b), we have the curve C defined as y = 4x. Similar to option (a), we evaluate two integrals: the first integral is with respect to x from 0 to 1, and the second integral is with respect to y from 0 to 4. The integrands for the x-component and y-component are the same as in option (a).

To find the specific numerical values of the line integrals, the integrals need to be solved using the given limits.

For C: y = 4x, I = ∫[0 to 1] (2.42 + 3 + 3²) dx + ∫[0 to 4] (4.2 - y) dy

= (2.42 + 3 + 3²) ∫[0 to 1] dx + ∫[0 to 4] (4.2 - 4x) dy

= (2.42 + 3 + 3²) [x] from 0 to 1 + (4.2y - xy) from 0 to 4

= (2.42 + 3 + 3²)(1 - 0) + (4.2(4) - (1)(4)) - (4.2(0) - (1)(0))

= (2.42 + 3 + 9)(1) + (16.8 - 4)

= (14.42)(1) + 12.8

= 14.42 + 12.8

= 27.22.

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1. If R is the area formed by the curve y = 5-x? dan y = (x - 1). Calculate the area R Dan = end

Answers

The area formed by the curves y = 5 - x and y = x - 1, denoted as R, can be calculated as 12 square units.

Determine the area?

To find the area formed by the two curves, we need to determine the points of intersection between them. By setting the two equations equal to each other, we can find the x-coordinate of the intersection point:

5 - x = x - 1

Simplifying the equation, we have:

2x = 6

x = 3

Substituting this x-coordinate back into either equation, we can find the corresponding y-coordinate:

y = 5 - x = 5 - 3 = 2

Therefore, the intersection point is (3, 2).

To calculate the area R, we integrate the difference between the two curves over the interval [3, 5] (the x-values where the curves intersect):

∫[3 to 5] [(5 - x) - (x - 1)] dx

Simplifying the expression, we have:

∫[3 to 5] (6 - 2x) dx

Integrating the function, we get:

[6x - x²] from 3 to 5

Substituting the limits of integration, we have:

[(6(5) - 5²) - (6(3) - 3²)]

Simplifying further, we get:

(30 - 25) - (18 - 9) = 5 - 9 = -4

However, since we are calculating the area, the value is positive, so the area R is 4 square units.

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Find the inverse Laplace transform of the following functions. 1 a) F(8) 2s + 3 32 - 4s + 3 QUESTION 2. Find the inverse Laplace transform of the following functions. 1 a) F(s) = 2s +3 s² - 4s +3

Answers

For the function F(s) = (2s + 3)/(32 - 4s + 3), the inverse Laplace transform can be directly obtained by evaluating F(s) at s = 8. For the function F(s) = (2s + 3)/(s^2 - 4s + 3), we need to first decompose it into partial fractions. Then, we can apply the inverse Laplace transform to each fraction to obtain the final solution.

1. F(8) = (2(8) + 3)/(32 - 4(8) + 3) = 19/27

2. To decompose F(s) into partial fractions, we write it as:

F(s) = A/(s-1) + B/(s-3)

To determine the values of A and B, we can multiply both sides by the denominators and equate the numerators:

(2s + 3) = A(s - 3) + B(s - 1)

Expanding and equating coefficients:

2s + 3 = (A + B)s + (-3A - B)

From here, we get a system of equations:

2 = A + B

3 = -3A - B

Solving this system, we find A = -1/2 and B = 5/2.

Therefore, the partial fraction decomposition of F(s) is:

F(s) = -1/2 * 1/(s - 1) + 5/2 * 1/(s - 3)

Now, we can take the inverse Laplace transform of each term using standard transform pairs:

L^-1 {1/(s - a)} = e^(at)

L^-1 {1/(s - b)} = e^(bt)

Applying these transforms, the inverse Laplace transform of F(s) becomes:

f(t) = -1/2 * e^t + 5/2 * e^(3t)

Therefore, the inverse transform of F(s) is given by f(t) = -1/2 * e^t + 5/2 * e^(3t).

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Other Questions
You have estimated the following Fama-French 3-factor model for Tesla stock (TSLA): Regression Model Outputs for TSLA Returns Coefficients Standard Error t-stat P-value Intercept 0.03 0.0235 1.0657 0.2911 Market 2.23 0.5305 4.2112 0.0001 SMB -0.42 0.8518 -0.4937 0.6235 HML -0.95 0.6482 -1.4686 0.1475 Assume that the risk-free rate is 1%. A. (1 point) What is the alpha of TSLA? Is it statistically significant at the 10%, 5%, or 1% threshold for significance? Briefly explain. B. (1 point) According to the estimated coefficients, which of the following styles represents TSLA: large-cap growth, large-cap value, small-cap growth, or small-cap value? Briefly explain. C. (1 point) Suppose that the expected market risk premium (MKT-RF) is 7%, SMB is 3%, and HML is 1%. What is the expected return on TSLA according to the Fama-French 3-factor model? most infants are physically ready to start eating solid foods when they .multiple choicelose the let-down reflexdevelop most of their primary teethhave kidneys that are physically matureare able to pick up foods with their fingers sleep is not a cessation of most brain activity but is thought to be important for cellular repair, brain development, and ______. a. circadian rhythm resets. b. homeostatic drive. c. synaptic repair. d. memory The cost of certain intangible assets are spread over their remaining legal or useful life, whichever is shorter. The expense to which this amount is recorded is called: Multiple Choice amortization expense depreciation expense. depletion expense. O deferred charge expense. researchers wanted to see if adults were actually afraid of the dark by exposing them to different levels of light in a room while measuring their heart rates. in this experiment, what are the different levels of light? an amount of 1.45 moles of carbon dioxide is produced when 5.50 moles of methane and 2.94 moles of oxygen are reaction in a 1.0 l container and 275 k. determine the equilibrium concentration of each substance and calculate the equilibrium constant Firm cvs has estimated sales (in millions) for the next four quarters as follows:Sales (in milions)Q1 Q2 Q3 Q470 100 120 130Sales for the first quarter of the year after this one are projected at $110 million. Accounts receivable at the beginning of the year were $23 milion. CVS has a 20,day collection period. CVS's purchases from suppliers in a quarter are equal to 30 percent of the next quarter's forecast sales, and suppliers are normally paidin 20 days. Wages, taxes, and other expenses run about 10 percent of sales. Interest and dividends are $7 milion per quarter.CVS plans a major capital outlay in the second quarter of $50 milion. Finally, the company started the year with a $35 milion cashbalance and wishes to maintain a $25 million minimum balance. Short term borrowing rate is 5%, short term lending rate is 3%.a-Prepare cash collection tableb- Prepare cash disbursement table c-Prepare cash budget tabled-Prepare short-term financial plan.e- What is the amount of cash generated by short-term financing? How might you use radioactive phosphorus to demonstrate that the transforming compound of bacteria in Griffith's experiment was DNA? [infinity] 1 Use the geometric series f(x): = = xk, for x < 1, to find the power series representation for the following 1-X k=0 function (centered at 0). Give the interval of convergence of the new series what is a disadvantage of multi-class dry chemical fire extinguishers ACB is bisected by segment CP. mWhat is the approximate area of ACB?NO SPAM OR I WILL REPORT YOU AND BAN YOU IMMEDIATELY Fritz Benjamin buys a car costing $18,600. He agrees to make payments at the end of each monthly period for 8 years. He pays 6.0% interest, compounded monthly (a) What is the amount of each payment? ( Which data suggest the strongest link between heredity and intelligence? If D is the triangle with vertices (0,0), (7,0), (7,20), then lloran D draw the molecular orbital diagram shown to determine which of the following is most stable. Robert borrows $45,000 from X bank. Jim, John and Keith agree to be co-sureties. The maximum liability for each of them is $9,000 for Jim, $13,500 for John, $22,500 for Keith. With $30,000 left to pay on the loan Robert defaults. If Keith pays $22,500 and Robert pays $7,500 how much can Keith get back from Jim and John? create an infinite geometric series to represent the decimal 0.44444... use this information to find the fraction to which this infinite geometric series converges. Zero-based budgeting is a unique technique for budgeting. It may work for some organizations but not for others.Complete an analysis of the zero-based approach to budgeting. Include the following in your analysis:Discuss the development of a decision package for existing and new programs and the ranking processIdentify an organization and discuss how the entity might use this approach effectively The demand curve of Lucky Egg in each district is shown as follow:0 = 1000 - 2P Suppose the manufacturer is the monopolist in the market of production. There are many distributors in the whole market but there is only one distributor ineach district (Each distributor is the monopolist in retail for a particular district). The marginal cost to produce a Lucky egg to the manufacturer is $100. The distribution cost to the distributor is $50 per egg. Determine the quantity transacted between one distributor and manufacturer in one district, quantity transacted between consumer and distributor in one district, the wholesale priceand the retail price respectively. Derivatives using Product Rule