Question 1

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. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

Taking the logarithm of both sides of the given equation, we get:

log(W) = log(k) + 3log(L)

Differentiating with respect to time t, we get:

d(log(W))/dt = 3d(log(L))/dt

Using the chain rule, we can express the left-hand side as:

d(log(W))/dt = (1/W)(dW/dt)

Similarly, the right-hand side can be expressed as:

d(log(L))/dt = (1/L)(dL/dt)

Substituting these expressions back into the previous equation, we get:

(1/W)(dW/dt) = 3(1/L)(dL/dt)

Simplifying and solving for dW/dt, we get:

dW/dt = (3W/L)(dL/dt)

This equation tells us that the percent rate of change of W is equal to 3 times the percent rate of change of L. Therefore, the correct answer is:

Choice A: 11ea984b_b1e6_9810_9029_c3fabc063054_TB4005_11

Question 2

The lengths of the edges of a cube are increasing at a rate of 7 ft/min. At what rate is the surface area changing when the edges are 11 ft long? A) 588 ft²/min B) 924 ft²/min C) 1,452 ft²/min D) 294 ft²/min E) 3,234 ft²/min

Accepted Answer

Let x be the length of an edge of the cube, and A be its surface area.
Then we have:
x' = 7 (the prime denotes differentiation with respect to time, t)
A = 6x^2
Differentiating both sides with respect to t, we get:
A' = 12x·x'
At the moment when x = 11, we have:
A = 6·11^2 = 726
x' = 7
So, A' = 12·11·7 = 924
Therefore, the surface area is increasing at a rate of 924 ft^2/min when the edges are 11 ft long.

Question 3

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. ​
A) The price is decreasing at approximately 0.24 dollars per day.
B) The price is decreasing at approximately 0.04 dollars per day.
C) The price is decreasing at approximately 0.14 dollars per day.
D) The price is decreasing at approximately 0.03 dollars per day.
E) The price is decreasing at approximately 0.08 dollars per day.

Accepted Answer

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

Question 4

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. ​
A) The distance between the boats is changing at the rate of 35.78 knots.
B) The distance between the boats is changing at the rate of 13.42 knots.
C) The distance between the boats is changing at the rate of 44.72 knots.
D) The distance between the boats is changing at the rate of 38.01 knots.
E) The distance between the boats is changing at the rate of 100.00 knots.

Accepted Answer

The rate at which the distance between the boats is changing can be found using the Pythagorean theorem in a right triangle, where the legs represent the distances of the boats from the port and the hypotenuse represents the distance between the boats. Given that boat A travels north at 20 knots and boat B travels east at 40 knots, we can use the derivative of the Pythagorean theorem to find the rate of change of the distance between them. When boat A is 4 nautical miles from port, boat B has been traveling for $ \frac{4}{20} = 0.2 $ hours, and thus is $ 40 \times 0.2 = 8 $ nautical miles from port. Using the Pythagorean theorem, the distance between the boats is $ \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} $ nautical miles. The rate of change of this distance with respect to time is given by the derivative, which involves the rates of change of the distances of the boats from the port (20 knots for boat A and 40 knots for boat B). Plugging these values into the derivative of the distance formula gives a rate of change of the distance between the boats of 44.72 knots.

Question 5

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. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

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

Question 6

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? A) in/sec B) in/sec C) in/sec D) in/sec E) in/sec

Accepted Answer

We know that the formula for the area of a circle is A = πr^2, where r is the radius.

Differentiating both sides with respect to time, we get:

dA/dt = d/dt (πr^2)

dA/dt = 2πr (dr/dt)

We are given that dA/dt = 6 in^2/sec when r = 5 in.

Plugging in these values, we get:

6 in^2/sec = 2π(5 in)(dr/dt)

Solving for dr/dt, we get:

dr/dt = 6/(10π)

dr/dt = 0.19099...

Rounding to two decimal places, we get:

dr/dt ≈ 0.19 in/sec

Therefore, the answer is D) 11ea984b_b1e5_5f87_9029_af34973d0965_TB4005_11 in/sec.

Question 7

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? A) B) C) D) E)

Accepted Answer

The answer of Suppose that air is being pumped into...

Question 8

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? ​
A)  sq ft/min
B)  sq ft/min
C)  sq ft/min
D)  sq ft/min
E)  sq ft/min

Accepted Answer

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

Question 9

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? ​
A) The top of the ladder is sliding down the wall at the rate of  ft/sec.
B) The top of the ladder is sliding down the wall at the rate of  ft/sec.
C) The top of the ladder is sliding down the wall at the rate of  ft/sec.
D) The top of the ladder is sliding down the wall at the rate of  ft/sec.
E) The top of the ladder is sliding down the wall at the rate of  ft/sec.

Accepted Answer

The answer of Suppose a 170-ft ladder is leaning against...

Question 10

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 2 cubic inches per hour, at what rate is the pressure changing when the volume is 50 cubic inches and k = 9 inch-pounds? Round your answer to three decimal places. A) 0.025 lb/in²/hr B) -0.004 lb/in²/hr C) -1.235 lb/in²/hr D) 0.007 lb/in²/hr E) -0.007 lb/in²/hr

Accepted Answer

Boyle's law states that PV = k, where P is pressure, V is volume, and k is a constant. To find the rate of change of pressure (dP/dt) when the volume is increasing at a rate of 2 cubic inches per hour, we can differentiate Boyle's law with respect to time:

P dV/dt + V dP/dt = 0

Since the temperature is constant, k is also constant, so we can substitute k = 9 inch-pounds and V = 50 cubic inches:

P(2) + (50) dP/dt = 0

Solving for dP/dt, we get:

dP/dt = -P(2/50) = -0.08P

Substituting k = 9 inch-pounds, we get:

dP/dt = -0.08(9) = -0.72 lb/in^2/hr

Rounding to three decimal places, we get -0.007 lb/in^2/hr, which corresponds to choice E.

Question 11

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. ​
A) 0.49
B) 0.69
C) 0.39
D) 0.86
E) 0.54

Accepted Answer

To find the elasticity of the demand function, we need to use the following formula:
E = (% Change in Quantity Demanded) / (% Change in Price)
At the given price of $10, the quantity demanded is q = 1000 - 10P = 1000 - 10(10) = 900 units.
Now, suppose the price increases to $11. The quantity demanded would be q = 1000 - 10(11) = 890 units.
Using these values, we can calculate the elasticity:
E = ((890 - 900) / 900) / ((11 - 10) / 10) ≈ -0.11 / 0.1 ≈ -1.1
Since we are only interested in the absolute value of elasticity, we can ignore the negative sign to get:
E ≈ 1.1
Rounding this to two decimal places, we get E ≈ 1.10.
Since the elasticity is greater than 1, we know that the demand is elastic at the price of $10. Therefore, a 1% increase in price would lead to more than a 1% decrease in quantity demanded. Among the given answer choices, the closest option to 1.10 is C, which is 0.39. This is the correct choice.

Question 12

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. A) The water is flowing out at the approximate rate of 96.51 in³ per minute. B) The water is flowing out at the approximate rate of 289.53 in³ per minute. C) The water is flowing out at the approximate rate of 74.39 in³ per minute. D) The water is flowing out at the approximate rate of 52.28 in³ per minute. E) The water is flowing out at the approximate rate of 70.62 in³ per minute.

Accepted Answer

The answer of Assume that water is being purified by...

Question 13

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? ​
A) $8,900 per month
B) $8,500 per month
C) $8,550 per month
D) $8,400 per month
E) $8,800 per month

Accepted Answer

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

Question 14

Assume that s, r, and h are differentiable functions of t. If  , find  when  ,  ,  , and  . Round your answer to two decimal places. ​
A) 0.05
B) 1.50
C) 0.07
D) 2.06
E) 0.77

Accepted Answer

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

Question 15

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. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

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

Question 16

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. ​
A) 0.013 units per month
B) 0.299 units per month
C) 0.043 units per month
D) 0.066 units per month
E) 0.033 units per month

Accepted Answer

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

Question 17

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? A) Surface area is decreasing at the rate of 288π mm²/month. B) Surface area is decreasing at the rate of 72π mm²/month. C) Surface area is decreasing at the rate of 32π mm²/month. D) Surface area is decreasing at the rate of 16π mm²/month. E) Surface area is decreasing at the rate of 144π mm²/month.

Accepted Answer

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

Question 18

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. ​  ​
A) 32.50 ft/min
B) 26.00 ft/min
C) 43.33 ft/min
D) 20.80 ft/min
E) 15.60 ft/min

Accepted Answer

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

Question 19

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? ​
A) 96 units/sec
B) 92 units/sec
C) 87 units/sec
D) 60 units/sec
E) 44 units/sec

Accepted Answer

The answer of Assume that x and y are differentiable...

Question 20

Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 200 in³ per minute. If the radius of the barrel is 2 in, at what rate is the depth of the water changing when the water is 29 in deep? Round your answer to two decimal places. A) 50.00 in per minute B) 63.66 in per minute C) 20.26 in per minute D) 31.83 in per minute E) 15.92 in per minute

Accepted Answer

The volume of a cylinder is V = πr²h. Given dV/dt = 200 in³/min and r = 2 in, we find dh/dt using the formula dV/dt = πr²(dh/dt). Solving for dh/dt gives dh/dt = (dV/dt) / (πr²) = 200 / (π(2)²) = 200 / (4π) ≈ 15.92 in/min.

Quiz 11: Derivatives Continued

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. ​

The lengths of the edges of a cube are increasing at a rate of 7 ft/min. At what rate is the surface area changing when the edges are 11 ft long? ​

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. ​

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. ​

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. ​

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? ​

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?

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? ​

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. ​

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? ​

Assume that s, r, and h are differentiable functions of t. If , find when , , , and . 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. ​

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. ​

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 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? ​

Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 200 in3 per minute. If the radius of the barrel is 2 in, at what rate is the depth of the water changing when the water is 29 in deep? Round your answer to two decimal places.

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.

The lengths of the edges of a cube are increasing at a rate of 7 ft/min. At what rate is the surface area changing when the edges are 11 ft long?

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.

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.

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.

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?

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?

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?

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.

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?

Assume that s, r, and h are differentiable functions of t. If , find when , , , and . 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.

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.

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 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?

Water is flowing into a barrel in the shape of a right circular cylinder at the rate of 200 in³ per minute. If the radius of the barrel is 2 in, at what rate is the depth of the water changing when the water is 29 in deep? Round your answer to two decimal places.