at what rate is the supply changing?

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 28000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of 8¢/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 = 28, solve the supply equation for p when x = 28.)

Answer 1

This is a business problem and I don't do too many business problems, but I think it is done this way.  I am sorry if this is not 100% correct. 

 

First solve for p. 

 

You get 625p²-x²=100

 

Plug in 28 for x to get

 

p=1.19

 

Implicitly differentiate (I did it below)

 

1250p *dp/dt -2x dx/dt=0

 

1250(1.19) (-.08) -2(28)dx/dt=0

 

 

14.28-56(dx/dt)=0

 

You get -56(dx/dt)=-14.28

 

dx/dt=-14.28/-56

 

dx/dt=.255

 

Multiply that by 1000 and it will be 255.