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 following equation.
$ 625p^2 - x^2 =100$

If 35000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of 5¢/carton/week, at what rate is the supply changing? (Round your answer to the nearest whole number.) (Hint: To find the value of p when x = 35, solve the supply equation for p when x = 35.)

___ cartons per week?

Answer 1

Answer:

1300

Step-by-step explanation:

We are given

x = 35

$\frac{dp}{dt}=-0.05$  [rate of change of price with time]

We want to find $\frac{dx}{dt}$, which is the rate at which supply is changing.

First, we need to implicitly differentiate the equation given:

$625p^2-x^2=100\\1250p\frac{dp}{dt}-2x\frac{dx}{dt}=0$

Here, we have what we need except for p. We find p when x = 35:

$625p^2 - x^2 =100\\625p^2 - (35)^2 =100\\625p^2=1325\\p=1.46$

Now, find dx/dt:

$1250(1.46)(-0.05)-2(35)\frac{dx}{dt}=0\\-91.25-70\frac{dx}{dt}=0\\\frac{dx}{dt}=-1.30$

In thousands, 1.30 * 1000 = 1300 cartons per week decrease