Question

The demand equation for a particular kind of shirt is 2px+65p-4950=0, where x hundreds of shirts are demanded per week when dollars is the price of a shirt. Suppose that the shirt is selling this week at $30, and the price increases at the rate of $0.20 per week. Find the rate of change in demand. 

Answer 1

$$2px+65p-4950=0$$

Differentiate both sides with respect to $t$

$$2p'x+2px'+65p'=0$$

Solve for $x'$

$$x'=-\dfrac{p'(2x+65)}{2p}$$

Given $p=30, p'=0.20$

$$2(30)x+65(30)-4950=0$$

$$x=50$$

Substitute

$$x'=-\dfrac{0.2(2(50)+65)}{2(30)}=-1.1$$

 The rate of change in demand is $-110$ shirts per week.