Question

Suppose that a certain market supplies x thousands of crates of orange daily when p dollars is the price per crate, and the supply equation is px-20p-3x+105=0. If the daily supply decreases at the rate of 250 crates per day, at what rate is the price changing when the daily supply is 5000 crates? 

Answer 1

$$px-20p-3x+105=0$$

Differentiate with respect to $t$

$$p'x+px'-20p'-3x'=0$$

Solve for $p'$

$$p'=\dfrac{(3-p)x'}{x-20}$$

Given $x'=-0.250, x=5$

$$p(5)-20p-3(5)+105=0$$

$$p=6$$

Substitute

$$p'=\dfrac{(3-6)(-0.25)}{5-20}=-0.05$$

The price is dropping by a nickel ($0.05).