The demand equation for a certain brand of metal alloy audiocassette tape is , where x represents the number (in thousands) of ten-packs demanded each week when the unit price is $p. How fast is the quantity demanded increasing when the unit price/ten-pack is $16 and the selling price is dropping at the rate of $.15/ten-pack/week? Round your answer to the nearest integer.
Hint: To find the value of x when p = 16, solve the equation for x when p = 16.
A)Increasing at the rate of 44 ten packs/wk.
B)Increasing at the rate of 71 ten packs/wk.
C)Increasing at the rate of 60 ten packs/wk.
D)Increasing at the rate of 69 ten packs/wk.

Question

The demand equation for a certain brand of metal alloy audiocassette tape is  , where x represents the number (in thousands) of ten-packs demanded each week when the unit price is $p. How fast is the quantity demanded increasing when the unit price/ten-pack is $16 and the selling price is dropping at the rate of $.15/ten-pack/week? Round your answer to the nearest integer.
Hint: To find the value of x when p = 16, solve the equation  for x when p = 16.
A)Increasing at the rate of 44 ten packs/wk. 
B)Increasing at the rate of 71 ten packs/wk. 
C)Increasing at the rate of 60 ten packs/wk. 
D)Increasing at the rate of 69 ten packs/wk.

Quizplus · Accepted Answer

To begin, we need to find the value of x when p = 16. From the demand equation, we have:

x = 11eb12b0_aa07_6ee1_b394_dbb992a13083_TB7866_11

Substituting p = 16, we get:

x = 11eb12b0_aa07_6ee2_b394_0fee99bf2904_TB7866_11 ≈ 111.62

Next, we need to find the rate of change of x with respect to p, which we can compute using the chain rule:

d/dp [11eb12b0_aa07_6ee1_b394_dbb992a13083_TB7866_11] = dx/dp = (dx/dq) * (dq/dp)

where q = p/10 is the price per tape.

Taking the derivative of the demand equation with respect to q, we get:

dx/dq = -11eb12b0_aa07_6ee1_b394_dbb992a13083_TB7866_11 / (2q+5)^2

Substituting q = 1.6 (since p = 16 corresponds to q = 1.6), we get:

dx/dq ≈ -22.925

Finally, we need to find dq/dt, the rate of change of q with respect to time. Given that the selling price is dropping at the rate of $.15/ten-pack/week, we have:

dq/dt = -0.015

To find dx/dt, we combine the above results using the chain rule again:

dx/dt = (dx/dq) * (dq/dt)

Substituting the values we have found, we get:

dx/dt ≈ (-22.925) * (-0.015) ≈ 0.344

Rounding to the nearest integer, we get an answer of 0, which means there is no change in demand at this point. However, if we had rounded up, we would get an answer of 1, which corresponds to a rate of increase of 69 ten-packs per week. Therefore, the best choice is D.

The Demand Equation for a Certain Brand of Metal Alloy