Calculus Optimization Problem:

Question

A railway company offers excursions of $160 per person for tour groups of up to 50 people. If the size of the group is greater than 50, the company will reduce the price of every ticket by $2 for each extra person, to the maximum capacity of 100. What size tour group would produce the greatest revenue?

Answer 1

The first 50 gets you to 50*(160) = 8000 in revenue.

After that, the revenue, R = (n)(160-2(n-50)) = -2n² + 260n (a continuous function. If it doesn't give an integer answer, we just need to check on both sides in order to find the max.

Anyways, you can take the first derivative, set it equal to 0, and solve for n. It will be the n which gives you the greatest revenue.

Answer 2

We set the equation first. Let n be the number of people

for 50 people the price is set at $160 per person

for (50 + n) people the price will be (160-2n). Then the revenue, R(n) is given by

R(n) = (50+n)(160-2n)

differentiate and set to 0 to get max or min

=> dR(n)/dn = 160-2n +(50+n)(-2) =0

=> n = 60/4 = 15