Eliminate the parameter / to rewrite the parametric equation as a Cartesian equation.
y (0) = t^5 +2
x(t) = -1

The marginal revenue of concession (g^) for the year 2018 is 7.59%.

To know marginal revenue of concession (g^) for the year 2018, we can use the following formula: [tex]g^1 = (Pt - Pt-1) / (Pt / (1 + Pt)),[/tex] Pt = Effective Price for the year t and Pt-1 = Effective Price for the previous year (t-1)

Using the given data, we will find the values of Pt and Pt-1 for the year 2018.

Pt = Effective Price for 2018-19 = $71.83

Pt-1 = Effective Price for 2017-18 = $66.53

Now, substituting values:

g^ = ($71.83 - $66.53) / ($71.83 / (1 + $71.83))

g^ = 0.0759

g^ = 7.59%.

Full question:

Year 2014-15 2015-16 2016-17 2017-18 2018-19 Avgs. NBA Data AvgTkt $53.98 $55.88 $58.67 $66.53 $71.83 $61.38 Attend/G 16,442 17,849 17,884 17,830 17,832 17568 FCI $333.58 $339.02 $355.97 $408.87 $420.65 g^ PT PE Marginal revenue of concession Profit maximizing price Effective Price (MRc + MRT) Ratio Ideal to Actual PT/P* g^ PE PT p"/p* 2015-16 2016-17 2017-18 2018-19 $55.88 $58.67 $66.53 $71.83. Calcuate the marginal revenue of concession (g^) for the year 2018.

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The present value of $4,500 received in two years at an interest rate of 7% is $3,928.51.

To calculate the present value of $4,500 received in two years at an interest rate of 7%, we need to use the present value formula, which is PV = FV / (1 + r) ^ n, where PV is the present value, FV is the future value, r is the interest rate, and n is the number of years.

So, in this case, we have FV = $4,500, r = 7%, and n = 2. Plugging these values into the formula, we get:

PV = $4,500 / (1 + 0.07) ^ 2
PV = $4,500 / 1.1449
PV = $3,928.51

This means that if you had $3,928.51 today and invested it at a 7% interest rate for two years, it would grow to $4,500 in two years.

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The area under the curve of f(x) = 4x + 8 between x = 6 and x = 8 is 96 square units.

The given function is f(x) = 4x + 8 and the interval is [6,8]. Using the Fundamental Theorem of Calculus, we can find the area under the curve of the function as follows:∫(from a to b) f(x)dx = F(b) - F(a)where F(x) is the antiderivative of f(x).The antiderivative of 4x + 8 is 2x^2 + 8x. Therefore,F(x) = 2x^2 + 8xNow, we can evaluate the area under the curve of f(x) as follows:∫[6,8] f(x)dx = F(8) - F(6) = [2(8)^2 + 8(8)] - [2(6)^2 + 8(6)] = 96

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To use Euler's method to approximate y(1) for the differential equation y' = y^2 - 2, with initial condition y(0) = 1, and a step size of h = 0.25.

We can use the following iterative formula:

y[i+1] = y[i] + h*f(x[i], y[i]), where f(x,y) = y^2 - 2, x[i] = i*h, and y[i] is the approximation of y at x = x[i].

Using this formula, we can approximate y at x = 1 as follows:

At i = 0: y[0] = 1

At i = 1:

x[1] = 0.25

f(x[0], y[0]) = (1)^2 - 2 = -1

y[1] = y[0] + hf(x[0], y[0]) = 1 + 0.25(-1) = 0.75

At i = 2:

x[2] = 0.5

f(x[1], y[1]) = (0.75)^2 - 2 ≈ -1.44

y[2] = y[1] + hf(x[1], y[1]) ≈ 0.75 + 0.25(-1.44) ≈ 0.39

Ati = 3:

x[3] = 0.75

f(x[2], y[2]) ≈ (0.39)^2 - 2 ≈ -1.98

y[3] = y[2] + hf(x[2], y[2]) ≈ 0.39 + 0.25(-1.98) ≈ 0.01

At i = 4:

x[4] = 1

f(x[3], y[3]) ≈ (0.01)^2 - 2 ≈ -1.9998

y[4] = y[3] + hf(x[3], y[3]) ≈ 0.01 + 0.25(-1.9998) ≈ -0.50

Therefore, using Euler's method with a step size of h = 0.25, we can approximate y(1) ≈ y[4] ≈ -0.50.

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To compute the limit as x approaches 0 of  [tex]\frac{5x^2}{cos(4x)-1}[/tex], we will utilize L'Hôpital's Rule. The limit evaluates to 5/8.

To compute the limit, we will apply L'Hôpital's Rule, which states that if the limit of a ratio of two functions exists in an indeterminate form (such as 0/0 or ∞/∞), then the limit of the ratio of their derivatives exists and is equal to the limit of the original function.

Let's evaluate the limit step by step:

lim (x->0)  [tex]\frac{5x^2}{cos(4x)-1}[/tex]

Since both the numerator and denominator approach 0 as x approaches 0, we have an indeterminate form of 0/0. Thus, we can apply L'Hôpital's Rule.

Taking the derivatives of the numerator and denominator:

lim (x->0) [tex]\frac{10x}{-4sin(4x)}[/tex]

Now we can evaluate the limit again:

lim (x->0) [tex]\frac{10x}{-4sin(4x)}[/tex]

Substituting x = 0 into the expression, we get:

lim (x->0) 0 / 0

Once again, we have an indeterminate form of 0/0. Applying L'Hôpital's Rule once more:

lim (x->0) [tex]\frac{10}{-16cos(4x)}[/tex]

Now we can evaluate the limit at x = 0:

lim (x->0)  [tex]\frac{10}{-16cos(4x)}[/tex] =  [tex]\frac{10}{-16cos(0)}[/tex] =  [tex]\frac{10}{-16(-1)}[/tex] = 10 / 16 = 5/8

Therefore, the limit as x approaches 0 of [tex]\frac{5x^2}{cos(4x)-1}[/tex] is 5/8.

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The correct question is:

Compute lim x->0   [tex]\frac{5x^2}{cos(4x)-1}[/tex]. Show each step, and state if you utilize l'Hôpital's Rule.

The sum of the geometric sequence in this problem is given as follows:

B) 166.7.

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.

The common ratio for this problem is given as follows:

q = 40/100

q = 0.4.

The formula for the sum of the infinite series is given as follows:

[tex]S = \frac{a_1}{1 - q}[/tex]

In which [tex]a_1[/tex] is the first term.

Hence the value of the sum is given as follows:

100/0.6 = 166.7.

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The limit of the radical expression [tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right)[/tex] as h approached 0 is 1/14

From the question, we have the following parameters that can be used in our computation:

[tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right)[/tex]

Rationalize the numerator in the above expression

So, we have the following representation

[tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right) = \lim _{h\to 0}\left(\frac{1}{\sqrt{49+h}+7}\right)[/tex]

Substitute 0 for h in the limit expression

So, we have

[tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right) = \left(\frac{1}{\sqrt{49+0}+7}\right)[/tex]

Evaluate the like terms

[tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right) = \left(\frac{1}{\sqrt{49}+7}\right)[/tex]

Take the square root of 49 and add to 7

[tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right) =\frac{1}{14}[/tex]

This means that the value of the limit expression is 1/14

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Question

Find the following limit or state that it does not exist.

[tex]\lim _{h\to 0}\left(\frac{\sqrt{49+h}-7}{h}\right)[/tex]

Answer:

-10

Step-by-step explanation:

-15 = 2x +5

move the numbers to one side

-15 + (-5) = 2x

-20 = 2x

devide by 2 to only be left with x

x = -10

To find the value of X in the equation -15 = 2X + 5, we can solve for X by isolating the variable on one side of the equation.

Given: -15 = 2X + 5

Subtracting 5 from both sides of the equation:

-15 - 5 = 2X + 5 - 5

-20 = 2X

To isolate X, we need to divide both sides of the equation by 2:

-20 / 2 = 2X / 2

-10 = X

Therefore, the value of X in the equation -15 = 2X + 5 is -10.

The probability of rolling two of the same number on a fair six-sided die is 1/6.

To calculate the probability of rolling two of the same number on a fair six-sided die, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of possible outcomes:

When rolling a fair six-sided die, there are six possible outcomes for each roll, as there are six faces on the die numbered 1 to 6.

Number of favorable outcomes:

To roll two of the same number, we can choose any number from 1 to 6 for the first roll.

The probability of rolling that number on the second roll to match the first roll is 1 out of 6, as there is only one favorable outcome.

This holds true for any number chosen for the first roll.

Therefore, there are 6 favorable outcomes, one for each number on the die.

Probability:

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Probability of rolling two of the same number = Number of favorable outcomes / Total number of possible outcomes

= 6 / 36

= 1 / 6

Thus, the probability of rolling two of the same number on a fair six-sided die is 1/6.

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The volume of a hemisphere with a radius of 44.9 m, rounded to the nearest tenth of a cubic meter, is approximately 222,232.7 cubic meters.

To calculate the volume of a hemisphere, we use the formula V = (2/3)πr³, where V represents the volume and r is the radius. In this case, the radius is 44.9 m. Plugging in the values, we get V = (2/3)π(44.9)³. Evaluating the expression, we find V ≈ 222,232.728 cubic meters. Rounding to the nearest tenth, the volume becomes 222,232.7 cubic meters.

The explanation of this calculation lies in the concept of a hemisphere. A hemisphere is a three-dimensional shape that is half of a sphere. The formula used to find its volume is derived from the formula for the volume of a sphere, but with a factor of 2/3 to account for its half-spherical nature. By substituting the given radius into the formula, we can find the volume. Rounding to the nearest tenth is done to provide a more precise and manageable value.

Therefore, the volume of a hemisphere with a radius of 44.9 m is approximately 222,232.7 cubic meters.

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It will take 5 seconds for the truck to cross the intersection from the moment the automobile is 100 meters away.

To solve this problem, we can use the concept of relative velocity. We'll consider the automobile as our reference point and calculate the relative velocity of the truck with respect to the automobile.

Given:

Speed of the automobile (v1) = 20 m/s

Distance of the automobile from the intersection (d1) = 100 meters

Speed of the truck (v2) = 40 m/s

We need to find the time it takes for the truck to cross the intersection from the moment the automobile is 100 meters away.

First, let's calculate the relative velocity of the truck with respect to the automobile:

Relative velocity (vrel) = v2 - v1

= 40 m/s - 20 m/s

= 20 m/s

Now, let's calculate the time it takes for the truck to cover the distance of 100 meters at the relative velocity:

Time (t) = Distance (d) / Relative velocity (vrel)

= 100 meters / 20 m/s

= 5 seconds

Therefore, it will take 5 seconds for the truck to cross the intersection from the moment the automobile is 100 meters away.

It's important to note that we assume both vehicles are moving in a straight line and maintaining a constant speed throughout the calculation. Additionally, we assume there are no external factors, such as acceleration or deceleration, that would affect the motion of the vehicles.

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The area of the region bounded by the function f(x), the Z-axis, and the vertical lines x = 2 and x = 3 are approximately XX square units.

To find the area of the region, we need to integrate the absolute value of the function f(x) over the given interval. Since f(x) is negative on (2, 3) and positive on (3, 4), we can split the integral into two parts.

First, we integrate the absolute value of f(x) over the interval (2, 3). The integral of f(x) over this interval will give us the negative area. Next, we integrate the absolute value of f(x) over the interval (3, 4), which will give us the positive area.

Adding the absolute values of these two areas will give us the total area of the region bounded by f(x), the Z-axis, and the vertical lines x = 2 and x = 3. Round the result to 2 decimal places.

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To find the position vector for the point that is halfway between point A(2, 7) and point B(14, 5), we can use the formula for the midpoint of two points.

The midpoint formula is given by: Midpoint = (1/2)(A + B), where A and B are the position vectors of the two points. Let's calculate the midpoint:

Midpoint = (1/2)(A + B) = (1/2)((2, 7) + (14, 5))

= (1/2)(16, 12)

= (8, 6). Therefore, the position vector for the point that is halfway between A(2, 7) and B(14, 5) is (8, 6). To find the position vector for the point that divides the line segment from A(2, 7) to B(14, 5) in the ratio 3:2, we can use the section formula.

The section formula is given by: Point = (rA + sB)/(r + s),where r and s are the ratios of the segment lengths. Let's calculate the position vector: Point = (3A + 2B)/(3 + 2) = (3(2, 7) + 2(14, 5))/(3 + 2)

= (6, 21) + (28, 10)/5

= (34, 31)/5

= (6.8, 6.2).Therefore, the position vector for the point that divides the line segment from A(2, 7) to B(14, 5) in the ratio 3:2 is approximately (6.8, 6.2).

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The probability that a student has been to the United Kingdom or Japan is 0.66.

This can be calculated using the following formula:

P(A or B) = P(A) + P(B) - P(A and B)

In this case, P(A) is the probability that a student has been to the United Kingdom, P(B) is the probability that a student has been to Japan, and P(A and B) is the probability that a student has been to both the United Kingdom and Japan.

Therefore, the probability that a student has been to the United Kingdom or Japan is:

P(A or B) = 0.28 + 0.52 - 0.14 = 0.66

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The total volume of the swimming pool is 160,000 cubic feet. A swimming pool is a man-made structure designed to hold water for recreational or competitive swimming activities.

To find the total volume of the swimming pool, we need to integrate the cross-sectional areas perpendicular to the x-axis over the entire length of the pool.

The equation of the ellipse representing the shape of the pool is given by:

(x^2/2500) + (y^2/400) = 1

To find the limits of integration, we need to determine the x-values where the ellipse intersects the x-axis. We can do this by setting y = 0 in the equation of the ellipse:

(x^2/2500) + (0^2/400) = 1

Simplifying, we get:

x^2/2500 = 1

x^2 = 2500

x = ±50

So, the ellipse intersects the x-axis at x = -50 and x = 50.

Now, we'll integrate the cross-sectional areas of the squares perpendicular to the x-axis. Since the cross sections are squares, the area of each cross section is equal to the side length squared.

For a given value of x, the side length of the square cross section is 2y, where y is given by the equation of the ellipse:

(y^2/400) = 1 - (x^2/2500)

Simplifying, we get:

y^2 = 400 - (400/2500)x^2

y = ±√(400 - (400/2500)x^2)

The cross-sectional area is then (2y)^2 = 4y^2.

To find the total volume, we integrate the cross-sectional areas from x = -50 to x = 50:

V = ∫[x=-50 to x=50] 4y^2 dx

V = 4∫[x=-50 to x=50] (√(400 - (400/2500)x^2))^2 dx

V = 4∫[x=-50 to x=50] (400 - (400/2500)x^2) dx

Simplifying and integrating, we get:

V = 4∫[x=-50 to x=50] (400 - (400/2500)x^2) dx

= 4[400x - (400/7500)x^3/3] |[x=-50 to x=50]

= 4[400(50) - (400/7500)(50)^3/3 - 400(-50) + (400/7500)(-50)^3/3]

= 4[20000 - (400/7500)(125000/3) + 20000 - (400/7500)(-125000/3)]

= 4[20000 - 666.6667 + 20000 + 666.6667]

= 4[40000]

= 160000

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To find an equation of a line that is tangent to the curve y = 5cos(2x) and whose slope is a minimum, we need to determine the derivative of the curve and set it equal to the slope of the tangent line. Then, we solve the resulting equation to find the x-coordinate(s) of the point(s) of tangency.

The derivative of y = 5cos(2x) can be found using the chain rule, which gives dy/dx = -10sin(2x). To find the slope of the tangent line, we set dy/dx equal to the desired minimum slope and solve for x: -10sin(2x) = minimum slope.

Next, we solve the equation -10sin(2x) = minimum slope to find the x-coordinate(s) of the point(s) of tangency. This can be done by taking the inverse sine of both sides and solving for x.

Once we have the x-coordinate(s), we substitute them back into the original curve equation y = 5cos(2x) to find the corresponding y-coordinate(s).

Finally, with the x and y coordinates of the point(s) of tangency, we can form the equation of the tangent line using the point-slope form of a line or the slope-intercept form.

In conclusion, by finding the derivative, setting it equal to the minimum slope, solving for x, substituting x into the original equation, and forming the equation of the tangent line, we can determine an equation of a line that is tangent to the curve y = 5cos(2x) and has a minimum slope.

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The answer is:A. The absolute maximum is at x = π/6, and the absolute minimums are at x = 5π/6 and x = 9π/6.

The given function is f(x) = sin 3x, and the given interval is [1, π]. We need to determine the location and value of the absolute extreme values of f(x) on the given interval, if they exist. Absolute extreme values refer to the maximum and minimum values of a function on a given interval. To find them, we need to find the critical points (where the derivative is zero or undefined) and the endpoints of the interval. We first take the derivative of f(x):f'(x) = 3cos 3xSetting this to zero, we get:3cos 3x = 0cos 3x = 0x = π/6, 5π/6, 9π/6 (or π/2)These are the critical points of the function. We then evaluate the function at the critical points and the endpoints of the interval: f(1) = sin 3 = 0.1411f(π) = sin 3π = 0f(π/6) = sin (π/2) = 1f(5π/6) = sin (5π/2) = -1f(9π/6) = sin (3π/2) = -1Therefore, the absolute maximum of the function on the given interval is 1, and it occurs at x = π/6. The absolute minimum of the function on the given interval is -1, and it occurs at x = 5π/6 and x = 9π/6.  

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The function A = 0.37 cos(t) + 0.45 models the amount of air (in liters) in an average resting person's lungs t seconds after exhaling.

The given function A = 0.37 cos(t) + 0.45 represents a mathematical model for the amount of air in liters in an average resting person's lungs t seconds after exhaling In the equation, cos(t) represents the cosine function, which oscillates between -1 and 1 as the input t varies. The coefficient 0.37 scales the amplitude of the cosine function, determining the range of values for the amount of air. The constant term 0.45 represents the average baseline level of air in the lungs.

The function A takes the input of time t in seconds and calculates the corresponding amount of air in liters. As t increases, the cosine function oscillates, causing the amount of air in the lungs to fluctuate around the baseline level of 0.45 liters. The amplitude of the oscillations is determined by the coefficient 0.37.

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a. The rocket will explode when the altitude reaches the value at which the atmospheric pressure, given by p(h) = 14.7e^(-h/10), drops below 10 pounds/sq in.

b. The rocket will explode if the atmospheric pressure drops below 10 pounds/sq in, as calculated by the height function y(t).

c. We need to determine the maximum height the rocket can reach before atmospheric pressure falls below 10 pounds/sq in.

a. To determine the altitude at which the rocket will explode, we need to find the value of h when p(h) = 14.7e^(-h/10) drops below 10. We set up the equation: 14.7e^(-h/10) = 10 and solve for h.

b. For x = 12 degrees, we can substitute this value into the height function y(t) = -16t^2 + t(1400sin(x)) and find the minimum height the rocket reaches. By converting the height to altitude, we can calculate the atmospheric pressure at that altitude using p(h) = 14.7e^(-h/10). If the pressure is below 10 pounds/sq in, the rocket will explode in midair.

c. To find the largest launch angle x so that the rocket will not explode, we need to determine the maximum height the rocket can reach before the atmospheric pressure falls below 10 pounds/sq in. This can be done by finding the value of x that maximizes the height function y(t) = -16t^2 + t(1400sin(x)). By setting the derivative of y(t) with respect to x equal to zero and solving for x, we can find the launch angle that ensures the rocket does not explode.

For a launch angle of x = 12 degrees, we can calculate the minimum atmospheric pressure exerted on the rocket. To find the largest launch angle x so that the rocket will not explode, we need to determine the maximum height the rocket can reach before the atmospheric pressure falls below 10 pounds/sq in by finding the value of x that maximizes the height function.

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The equation of the plane is:5x + 8y + 8z = 29 In terms of the variables x, y, and z, the equation of the plane is 5x + 8y + 8z = 29.

To find an equation of the plane through the point (1, 5, -2) with a normal vector (5, 8, 8), we can use the general equation of a plane:

Ax + By + Cz = D

where (A, B, C) is the normal vector of the plane and (x, y, z) are the coordinates of any point on the plane.

Given the normal vector (5, 8, 8) and the point (1, 5, -2), we can substitute these values into the equation and solve for D:

5x + 8y + 8z = D

Plugging in the coordinates (1, 5, -2):

5(1) + 8(5) + 8(-2) = D

5 + 40 - 16 = D

29 = D

Therefore, the equation of the plane is:

5x + 8y + 8z = 29

In terms of the variables x, y, and z, the equation of the plane is 5x + 8y + 8z = 29.

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The cost of the cylinder in terms of the single variable, r, alone is 2000π + πr⁴

The volume of a cylinder is given by πr²h. We know that the volume of the cylinder must be 1000π cubic feet, so we can set up the following equation:

πr²h = 1000π

h = 1000/r²

The cost of the cylinder is given by 2πr²h + πr² = 2πr²(1000/r²) + πr² = 2000π + πr⁴

The cost of the cylinder in terms of the single variable, r, alone is:

Cost of cylinder = 2000π + πr⁴

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The volume of the solid obtained by rotating the region bounded by the curves y = 2x², y = 2x, and the x-axis, about the x-axis, is (32π/15) cubic units.

To find the volume of the solid, we can use the method of cylindrical shells. The solid is formed by rotating a vertical strip bounded by the curves about the x-axis.

The height of each cylindrical shell is the difference between the y-values of the upper and lower curves, which is (2x - 2x²).

The radius of each shell is the x-coordinate at which the curves intersect, which can be found by equating the two equations: 2x = 2x².

Solving this equation, we find two intersection points at x = 0 and x = 1.

Using the formula for the volume of a cylindrical shell, V = ∫(2πrh)dx, where r is the radius and h is the height, we integrate from x = 0 to x = 1. Substituting the values of r = x and h = (2x - 2x²), the integral becomes V = ∫(2πx(2x - 2x²))dx.

Simplifying the integral, we obtain V = (32π/15) cubic units. Therefore, the volume of the solid obtained by rotating the given region about the x-axis is (32π/15) cubic units.

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The work done by the force F = (x sin y, y) along the curve y = r^2 from (-1, 1) to (2, 4) is 18.1089.

Step 1: Parameterize the curve:

Since the curve is defined by y = r^2, we can parameterize it as r(t) = (t, t^2), where t varies from -1 to 2.

Step 2: Calculate dr:

To find the differential displacement dr along the curve, we differentiate the parameterization with respect to t: dr = (dt, 2t dt).

Step 3: Substitute into the line integral formula:

The work done by the force F along the curve can be expressed as the line integral:

W = ∫C F · dr,

where F = (x sin y, y) and dr = (dt, 2t dt). Substituting these values:

W = ∫C (x sin y, y) · (dt, 2t dt).

Step 4: Evaluate the dot product:

The dot product (x sin y, y) · (dt, 2t dt) is given by (x sin y) dt + 2ty dt.

Step 5: Express x and y in terms of the parameter t:

Since x is simply t and y is t^2 based on the parameterization, we have:

(x sin y) dt + 2ty dt = (t sin (t^2)) dt + 2t(t^2) dt.

Step 6: Integrate over the given range:

Now, we integrate the expression with respect to t over the range -1 to 2:

W = ∫[-1 to 2] (t sin (t^2)) dt + ∫[-1 to 2] 2t(t^2) dt.

Step 7: Evaluate the integrals:

Using appropriate techniques to evaluate the integrals, we find that the first integral equals approximately -0.0914, and the second integral equals 18.2003.

Therefore, the work done by the force F along the curve y = r^2 from (-1, 1) to (2, 4) is approximately 18.1089 (rounded to four decimal places).

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The Gram-Schmidt process is used to transform the basis {u₁, u₂, u₃, u₄} into an orthonormal basis {v₁, v₂, v₃, v₄} in R⁴.


The Gram-Schmidt process is a method used to transform a given basis into an orthonormal basis by orthogonalizing and normalizing the vectors. In this case, we are working in R⁴ with the basis {u₁, u₂, u₃, u₄}, where u₁ = (1, 0, 0, 0) and u₂ = (1, 1, 0, 0).

To apply the Gram-Schmidt process, we start by setting v₁ = u₁ and normalize it to obtain the first orthonormal vector. Since u₁ is already normalized, v₁ remains unchanged.

Next, we orthogonalize u₂ with respect to v₁. We subtract the projection of u₂ onto v₁ from u₂ to obtain a vector orthogonal to v₁. Let's call this new vector w₂. Then, we normalize w₂ to obtain v₂, the second orthonormal vector.

Continuing the process, we orthogonalize u₃ with respect to v₁ and v₂, and then normalize the resulting vector to obtain v₃, the third orthonormal vector.

Finally, we orthogonalize u₄ with respect to v₁, v₂, and v₃, and normalize the resulting vector to obtain v₄, the fourth and final orthonormal vector.

The resulting orthonormal basis is {v₁, v₂, v₃, v₄}, where each vector is orthogonal to the previous ones and has a length of 1, representing an orthonormal basis in R⁴.

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To write the region of integration D in rectangular coordinates, we need to determine the bounds for x, y, and z.

From the given limits of integration, we have:

[tex]-4 ≤ x ≤ 0[/tex]

[tex]0 ≤ y ≤ √(16 - x^2)[/tex]

[tex]0 ≤ z ≤ x^2 + y^2[/tex]

Therefore, the region of integration D in rectangular coordinates is:

[tex]D: -4 ≤ x ≤ 0, 0 ≤ y ≤ √(16 - x^2), 0 ≤ z ≤ x^2 + y^2.[/tex]

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The value of k that divides the area enclosed by the curves y=x, y=0, and x=5 into equal parts is approximately 3.54.

To find this value, we need to calculate the area enclosed by the given curves between x=0 and x=5, and then determine the point where the area is divided equally.

The area enclosed by the curves is given by the integral of y=x from x=0 to x=5. Integrating y=x with respect to x gives us the area as [tex](1/2)x^2.[/tex]

Next, we set up an equation to find the value of k where the area is divided equally. We can write the equation as follows: [tex](1/2)k^2 = (1/2)(5^2 - k^2).[/tex]Solving this equation, we find that k ≈ 3.54.

Therefore, the vertical line x=3.54 divides the area enclosed by the curves y=x, y=0, and x=5 into equal parts.

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Answer:

It was snowing for 4 hours and 23 minutes

Step-by-step explanation:

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4 hours and 23 minutes.

A rectangular R region with (1,2) , (2,3) , (3,2) and(2,1) as corners, then the value of the integral over R is 3 ln 3 - 2 using their limits of integration.

To evaluate the integral ∬_R ln(x+y) dA over the rectangular region R with corners (1,2), (2,3), (3,2), and (2,1), we can use the change of variables u = x + y and v = x - y. This transformation maps the region R to a parallelogram P with vertices at (3,1), (4,1), (3,4), and (2,4).

The Jacobian of this transformation is:

| ∂u/∂x  ∂u/∂y |

| ∂v/∂x  ∂v/∂y | = | 1 1 |

                            | 1 -1 | = -2

Therefore, the integral becomes:

∬_P ln(u)/|-2| dA

where u = x+y and v=x-y. Solving for x and y in terms of u and v, we get:

x = (u+v)/2

y = (u-v)/2

The limits of integration for u and v are determined by the vertices of the parallelogram P:

1 ≤ x-y ≤ 2    -->    -1 ≤ v ≤ 0

1 ≤ x+y ≤ 3    -->    1 ≤ u ≤ 3

3 ≤ x-y ≤ 4    -->    1 ≤ v ≤ 2

2 ≤ x+y ≤ 4    -->    3 ≤ u ≤ 4

Therefore, the integral becomes:

∬_P ln(u)/2 dA

= (1/2) ∫_1^3 ∫_{-u+1}^{u-1} ln(u) dv du + (1/2) ∫_3^4 ∫_{u-2}^{2-u} ln(u) dv du

= (1/2) ∫_1^3 [ln(u)(2-u+1-u)] du + (1/2) ∫_3^4 [ln(u)(2u-2u)] du

= (1/2) ∫_1^3 2ln(u) du

= ∫_1^3 ln(u) du

= [u ln(u) - u]_1^3

= 3 ln 3 - 2

Therefore, the value of the integral over R is 3 ln 3 - 2.

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Given series is Σ 4n3 + 2 n=1.  if the limit of [tex]a_n[/tex] is not equal to zero or if the limit does not exist, then the series is divergent.

We need to check whether the given series converges or diverges. Divergence test states that if the limit of a series is not zero, then the series is divergent.

In the given series, 4n3 is an increasing function as value of n increases. Therefore, it is not possible for the limit to be zero. Hence, we can say that the given series does not converge.Based on Divergence Test, the given series diverges. Therefore, the correct option is O Diverges.

Note: The Divergence Test is a simple test that says, if an infinite series [tex]a_n[/tex] is such that lim [tex]a_n[/tex]≠ 0, then the series does not converge and is said to diverge. In other words, if the limit of [tex]a_n[/tex] is not equal to zero or if the limit does not exist, then the series is divergent.

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We can state that the value οf the definite integral ∫₀³ x³ dx is between 0 and 81.

smaller value = 0

larger value = 81

Tο estimate the value οf the definite integral ∫₀³ x³ dx using the given prοperty, we need tο find the absοlute minimum and maximum οf the functiοn f(x) = x³ οn the interval [0, 3].

Taking the derivative οf f(x) and setting it tο zerο tο find critical pοints:

f'(x) = 3x²

3x² = 0

x = 0

We have a critical pοint at x = 0.

Nοw let's evaluate the functiοn at the critical pοint and the endpοints οf the interval:

f(0) = 0³ = 0

f(3) = 3³ = 27

Frοm the abοve calculatiοns, we can see that the absοlute minimum (m) οf f(x) οn the interval [0, 3] is 0, and the absοlute maximum (M) is 27.

Nοw we can use the given prοperty tο estimate the value οf the definite integral:

m(b - a) ≤ ∫₀³ x³ dx ≤ M(b - a)

0(3 - 0) ≤ ∫₀³ x³ dx ≤ 27(3 - 0)

0 ≤ ∫₀³ x³ dx ≤ 81

Therefοre, we can estimate that the value οf the definite integral ∫₀³ x³ dx is between 0 and 81.

smaller value = 0

larger value = 81

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