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 $20 and the selling price is dropping at the rate of $.11/ten-pack/week? Round your answer to the nearest integer.
Hint: To find the value of x when p = 20, solve the equation for x when p = 20.
A) Increasing at the rate of 93 ten packs/wk.
B) Increasing at the rate of 82 ten packs/wk.
C) Increasing at the rate of 66 ten packs/wk.
D) Increasing at the rate of 31 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 $20 and the selling price is dropping at the rate of $.11/ten-pack/week? Round your answer to the nearest integer.
Hint: To find the value of x when p = 20, solve the equation  for x when p = 20.
A) Increasing at the rate of 93 ten packs/wk. 
B) Increasing at the rate of 82 ten packs/wk. 
C) Increasing at the rate of 66 ten packs/wk. 
D) Increasing at the rate of 31 ten packs/wk. ​

Quizplus · Accepted Answer

To find the value of x when p = 20, we substitute p = 20 in the demand equation and solve for x:
11x = 220 - 20^2
11x = 20
x ≈ 18.2
Now we need to find the rate at which the quantity demanded is increasing, which is given by:
d/dt (11x) = 11(dx/dt)
We are given that dp/dt = -0.11, so we can use the chain rule to find dx/dt:
dx/dt = (dx/dp)(dp/dt) = (-1/11p)(-0.11) = 0.01p
At p = 20, dx/dt = 0.01(20) = 0.2, or approximately 0.2 thousand ten-packs per week. Rounding to the nearest integer, we get an increase in demand of 66 ten-packs per week. Therefore, the answer is C.

The Demand Equation for a Certain Brand of Metal Alloy