The demand function for a bottle of hand lotion is p(x)=0.78+0.0003x, where x is the number of bottles sold and p is the price, in dollars. The cost function is c(x)=480-0.32x+0.0005x^2. Find the marginal profit when 700 bottles of hand lotion are sold.

Answer 1

profit = revenue - cost
= demand * price - cost
= x*p(x)-c(x)
= x(.78+.003x) - (480-.32x+.0005x^2)
= 0.0025x^2 + 1.1x - 480

marginal profit = dp/dx = 0.005x^2 + 1.1
p'(700) = 0.005*700^2 + 1.1 = 2451.1