A theme park sells tickets at a price of $150 per person. After 5,000 people have bought entrance into the park, however, the park reduces the ticket price by $0.50 for every extra person.

Question

Using calculus skills, find the number of entrants that maximizes the revenue received at the theme park. And show your work step by step

Answer 1

Let x be the extra people over 5000.

Revenue = quantity time price

Revenue, R(x) = 5000(150)+x(150-0.5x) =750000 +150x-x²/2

For every person over 5000, the price is reduced by another $0.50.

To maximize, find R'(x), set that to zero and solve for x.

R'(x) = 150-x

150-x=0

x=150

Revenue is maximized at 5000+150 = 5150 entrants.