Question 1

The area of a circle is changing at a rate of 6 in²/sec At what rate is its radius changing when the radius is 5 in?

Accepted Answer

The answer of The area of a circle is changing...

Question 2

The radius of a circle is increasing at a rate of 6 ft/min. At what rate is its area changing when the radius is 7 ft? ​

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The answer of The radius of a circle is increasing...

Question 3

Suppose a 170-ft ladder is leaning against a wall. If the bottom is pulled away from the wall at a rate of 1 ft/sec, at what rate is the top of the ladder sliding down the wall when the bottom is 26 ft from the wall? ​

Accepted Answer

To solve this problem, we need to use related rates. Let's label the sides of the right triangle formed by the wall, the ground, and the ladder.

- Let x be the distance from the ladder's base on the ground to the wall.
- Let y be the height of the ladder on the wall.
- Let z be the length of the ladder (which we know is 170 ft).

From the Pythagorean theorem, we know that:

x^2 + y^2 = z^2

Now, we'll implicitly differentiate this equation with respect to time (t) using the chain rule:

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

Note that we don't know dx/dt or dy/dt, but we do know that dx/dt = -1 (since we're told that the bottom of the ladder is being pulled away from the wall at a rate of 1 ft/sec). We're also given that z = 170 ft. Finally, we need to find dy/dt when x = 26 ft.

We can plug in all our known values and solve for dy/dt:

2(26)(-1) + 2y(dy/dt) = 2(170)(0)

-52 + 2y(dy/dt) = 0

2y(dy/dt) = 52

dy/dt = 52/(2y)

At this point, we need to find y when x = 26. We can use the Pythagorean theorem to do this:

26^2 + y^2 = 170^2

y^2 = 170^2 - 26^2 = 27,446

y = sqrt(27,446) = 165.6 (rounded to one decimal place)

Now we can plug in y and solve for the rate at which y is changing (i.e. dy/dt):

dy/dt = 52/(2y) = 52/(2*165.6)

dy/dt = 0.157 ft/sec

So the top of the ladder is sliding down the wall at a rate of approximately 0.157 ft/sec when the bottom is 26 ft from the wall.

Therefore, the best choice is C.

Question 4

Assume that x and y are differentiable functions of t. A point is moving along the graph of the equation  At what rate is y changing when  and is changing at a rate of 4 units/sec? ​

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Question 5

Suppose that air is being pumped into a spherical balloon at a rate of 12 in³/min.At what rate is the radius of the balloon increasing when the radius is 9 in?

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Question 6

Suppose that a boat is being pulled toward a dock by a winch that is 24 ft above the level of the boat deck. If the winch is pulling the cable at a rate of 26 ft/min, at what rate is the boat approaching the dock when it is 32 ft from the dock? Use the figure below. Round your answer to two decimal places. ​  ​

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The answer of Suppose that a boat is being pulled...

Question 7

Assume that water is being purified by causing it to flow through a conical filter that has a height of 5 inches and a radius of 4 inches. If the depth of the water is decreasing at a rate of 3 inches per minute when the depth is 4 inches, at what rate is the volume of water flowing out of the filter at this instant? Round your answer to two decimal places. ​

Accepted Answer

The volume of a cone is given by the formula V = (1/3)πr^2h. If the depth of the water is decreasing at a rate of 3 inches per minute when the depth is 4 inches, then the radius of the water is also decreasing at a rate of 3 inches per minute. So, the rate of change of the volume of water is given by:dV/dt = (1/3)π(2rh)(dr/dt) + (1/3)πr^2(dh/dt)= (1/3)π(2(4)(3)) + (1/3)π(4)^2(-3)= 96.51 in^3/minTherefore, the water is flowing out at the approximate rate of 96.51 in^3 per minute.

Question 8

Suppose that the monthly revenue and cost (in dollars) for x units of a product are  . At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? ​

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The answer of Suppose that the monthly revenue and cost...

Question 9

Boyle's law for enclosed gases states that at a constant temperature, the pressure is related to the volume by the equation  , where k is a constant. If the volume is increasing at a rate of 7 cubic inches per hour, at what rate is the pressure changing when the volume is 40 cubic inches and  inch-pounds? Round your answer to three decimal places.

Accepted Answer

From the given equation 11eb0d3b_9883_3849_9790_3fe83bde04dd_TB1243_11 , we can take the derivative with respect to time to find the rate of change of pressure with respect to volume: dP/dt = -k/V^2 * dV/dt Plugging in the given values, we get: dP/dt = -(11eb0d3b_9883_384a_9790_61b5b19d4e63_TB1243_11)/(40)^2 * 7 Simplifying, we get: dP/dt = -0.009 lb/in²/hr Therefore, the answer is E.

Question 10

The supply function for a product is given by  , where x is the number of units supplied and p is the price in dollars. If the price is increasing at a rate of $1 per month, at what rate is the supply changing when  Round your answer to three decimal places. ​

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The answer of The supply function for a product is...

Question 11

Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 700 in³ per minute. If the radius of the barrel is 3 in, at what rate is the depth of the water changing when the water is 10 in deep? Round your answer to two decimal places.

Accepted Answer

The volume of a cylinder is V = πr²h. Given dV/dt = 700 in³/min and r = 3 in, we find dh/dt using dV/dt = πr²(dh/dt). Solving for dh/dt gives dh/dt = 700/(π*3²) ≈ 24.76 in/min.

Question 12

Suppose that the price p (in dollars) of a product is given by the demand function  where x represents the quantity demanded. If the daily demand is increase at a rate of 5 units per day, at what rate is the price changing when the demand is 64 units? Round your answer to two decimal places. ​

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The answer of Suppose that the price p (in dollars)...

Question 13

The resistance R of a blood vessel to the flow of blood is a function of the radius r of the blood vessel and is given by  , where k is a constant. For  find the percent rate of change of the resistance of a blood vessel in terms of the percent rate of change in the radius of the blood vessel. ​

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The answer of The resistance R of a blood vessel...

Question 14

Two boats leave the same port at the same time, with boat A traveling north at 20 knots (nautical miles per hour) and boat B traveling east at 40 knots. How fast is the distance between them changing when boat A is 4 nautical miles from port? Round your answer to two decimal places. ​

Accepted Answer

The distance between the boats can be found using the Pythagorean theorem, $d = \sqrt{x^2 + y^2}$, where $x$ and $y$ are the distances of the boats from the port along the east and north directions, respectively. Given $y = 4$ nautical miles (for boat A) and the rates $dx/dt = 40$ knots (for boat B) and $dy/dt = 20$ knots (for boat A), we can find $dd/dt$ using differentiation: $dd/dt = (1/2)(2x dx/dt + 2y dy/dt) / \sqrt{x^2 + y^2}$. At the time when $y = 4$, assuming $x = 8$ (since boat B travels twice as fast as boat A), we find $dd/dt = (1/2)(2*8*40 + 2*4*20) / \sqrt{8^2 + 4^2} = 44.72$ knots.

Question 15

p is in dollars and q is the number of units. Find the elasticity of the demand function  at the price p = $10. Round your answer to two decimal places. ​

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The answer of p is in dollars and q is...

Question 16

For human beings, the surface area S of the body is related to the body's weight W according to  , where k is a constant. For  , find the percent rate of change of the body's surface area in terms of the percent rate of change of the body's weight. ​

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The answer of For human beings, the surface area S...

Question 17

For many species of fish, the allometric relationship between the weight W and the length L is approximately  where k is a constant. For  find the percent rate of change of the weight as a corresponding percent rate of change of the length. ​

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The answer of For many species of fish, the allometric...

Question 18

Suppose that a tumor in a person's body has a spherical shape and that treatment is causing the radius of the tumor to decrease at a rate of 9 millimeters per month. At what rate is the surface area of the tumor decreasing when the radius is 2 mm? ​

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The answer of Suppose that a tumor in a person's...

Question 19

Assume that s, r, and h are differentiable functions of t. If  , find  when  ,  ,  , and  . Round your answer to two decimal places.

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The answer of Assume that s, r, and h are...

Question 20

The lengths of the edges of a cube are increasing at a rate of 9 ft/min. At what rate is the surface area changing when the edges are 16 ft long? ​

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The answer of The lengths of the edges of a...

Quiz 12: Derivatives Continued

The area of a circle is changing at a rate of 6 in2/sec At what rate is its radius changing when the radius is 5 in? ​

The radius of a circle is increasing at a rate of 6 ft/min. At what rate is its area changing when the radius is 7 ft? ​

Suppose a 170-ft ladder is leaning against a wall. If the bottom is pulled away from the wall at a rate of 1 ft/sec, at what rate is the top of the ladder sliding down the wall when the bottom is 26 ft from the wall? ​

Assume that x and y are differentiable functions of t. A point is moving along the graph of the equation At what rate is y changing when and is changing at a rate of 4 units/sec? ​

Suppose that air is being pumped into a spherical balloon at a rate of 12 in3/min.At what rate is the radius of the balloon increasing when the radius is 9 in?

Suppose that the monthly revenue and cost (in dollars) for x units of a product are . At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? ​

The supply function for a product is given by , where x is the number of units supplied and p is the price in dollars. If the price is increasing at a rate of $1 per month, at what rate is the supply changing when Round your answer to three decimal places. ​

Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 700 in3 per minute. If the radius of the barrel is 3 in, at what rate is the depth of the water changing when the water is 10 in deep? Round your answer to two decimal places.

Suppose that the price p (in dollars) of a product is given by the demand function where x represents the quantity demanded. If the daily demand is increase at a rate of 5 units per day, at what rate is the price changing when the demand is 64 units? Round your answer to two decimal places. ​

The resistance R of a blood vessel to the flow of blood is a function of the radius r of the blood vessel and is given by , where k is a constant. For find the percent rate of change of the resistance of a blood vessel in terms of the percent rate of change in the radius of the blood vessel. ​

Two boats leave the same port at the same time, with boat A traveling north at 20 knots (nautical miles per hour) and boat B traveling east at 40 knots. How fast is the distance between them changing when boat A is 4 nautical miles from port? Round your answer to two decimal places. ​

p is in dollars and q is the number of units. Find the elasticity of the demand function at the price p = $10. Round your answer to two decimal places. ​

For human beings, the surface area S of the body is related to the body's weight W according to , where k is a constant. For , find the percent rate of change of the body's surface area in terms of the percent rate of change of the body's weight. ​

For many species of fish, the allometric relationship between the weight W and the length L is approximately where k is a constant. For find the percent rate of change of the weight as a corresponding percent rate of change of the length. ​

Suppose that a tumor in a person's body has a spherical shape and that treatment is causing the radius of the tumor to decrease at a rate of 9 millimeters per month. At what rate is the surface area of the tumor decreasing when the radius is 2 mm? ​

Assume that s, r, and h are differentiable functions of t. If , find when , , , and . Round your answer to two decimal places.

The lengths of the edges of a cube are increasing at a rate of 9 ft/min. At what rate is the surface area changing when the edges are 16 ft long? ​

The area of a circle is changing at a rate of 6 in²/sec At what rate is its radius changing when the radius is 5 in?

The radius of a circle is increasing at a rate of 6 ft/min. At what rate is its area changing when the radius is 7 ft?

Suppose a 170-ft ladder is leaning against a wall. If the bottom is pulled away from the wall at a rate of 1 ft/sec, at what rate is the top of the ladder sliding down the wall when the bottom is 26 ft from the wall?

Assume that x and y are differentiable functions of t. A point is moving along the graph of the equation At what rate is y changing when and is changing at a rate of 4 units/sec?

Suppose that air is being pumped into a spherical balloon at a rate of 12 in³/min.At what rate is the radius of the balloon increasing when the radius is 9 in?

Suppose that the monthly revenue and cost (in dollars) for x units of a product are . At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month?

The supply function for a product is given by , where x is the number of units supplied and p is the price in dollars. If the price is increasing at a rate of $1 per month, at what rate is the supply changing when Round your answer to three decimal places.

Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 700 in³ per minute. If the radius of the barrel is 3 in, at what rate is the depth of the water changing when the water is 10 in deep? Round your answer to two decimal places.

Suppose that the price p (in dollars) of a product is given by the demand function where x represents the quantity demanded. If the daily demand is increase at a rate of 5 units per day, at what rate is the price changing when the demand is 64 units? Round your answer to two decimal places.

The resistance R of a blood vessel to the flow of blood is a function of the radius r of the blood vessel and is given by , where k is a constant. For find the percent rate of change of the resistance of a blood vessel in terms of the percent rate of change in the radius of the blood vessel.

Two boats leave the same port at the same time, with boat A traveling north at 20 knots (nautical miles per hour) and boat B traveling east at 40 knots. How fast is the distance between them changing when boat A is 4 nautical miles from port? Round your answer to two decimal places.

p is in dollars and q is the number of units. Find the elasticity of the demand function at the price p = $10. Round your answer to two decimal places.

For human beings, the surface area S of the body is related to the body's weight W according to , where k is a constant. For , find the percent rate of change of the body's surface area in terms of the percent rate of change of the body's weight.

For many species of fish, the allometric relationship between the weight W and the length L is approximately where k is a constant. For find the percent rate of change of the weight as a corresponding percent rate of change of the length.

Suppose that a tumor in a person's body has a spherical shape and that treatment is causing the radius of the tumor to decrease at a rate of 9 millimeters per month. At what rate is the surface area of the tumor decreasing when the radius is 2 mm?

The lengths of the edges of a cube are increasing at a rate of 9 ft/min. At what rate is the surface area changing when the edges are 16 ft long?