Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the equation . If 23,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 800 cartons/week, at what rate is the wholesale price changing? (Hint: To find the value of p when , solve the supply equation for p when ).
A) Increasing at the rate of
B) Increasing at the rate of
C) Dropping at the rate of
D) Dropping at the rate of
E) Dropping at the rate of

Question

Suppose the wholesale price of a certain brand of medium-sized eggs p (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the equation  . If 23,000 cartons of eggs are available at the beginning of a certain week and the supply is falling at the rate of 800 cartons/week, at what rate is the wholesale price changing? (Hint: To find the value of p when  , solve the supply equation for p when  ).
A) Increasing at the rate of 
B) Increasing at the rate of 
C) Dropping at the rate of 
D) Dropping at the rate of 
E) Dropping at the rate of

Quizplus · Accepted Answer

To find the rate at which the wholesale price is changing, we need to take the derivative of the equation with respect to time. Using the product rule, we have:

dp/dt = (dp/dx) (dx/dt)

We are given that dx/dt = -800 (since the supply is falling at a rate of 800 cartons/week). To find dp/dx, we first need to solve the supply equation for p:

p = 1200 - 0.03x

Taking the derivative of this equation with respect to x, we get:

dp/dx = -0.03

Substituting this and dx/dt into the rate of change equation, we get:

dp/dt = (-0.03) (-800) = 24

Therefore, the wholesale price is dropping at a rate of $24/carton/week. So the correct answer is C.

Suppose the Wholesale Price of a Certain Brand of Medium-Sized