Optimization Calculus:

A bus company offers excursions at $120 per person for tour groups of up to 100 persons. If the size of a group is greater than 100, the company will reduce the price of every ticket by 50 cents for every person in excess of 100. What size of a tour group would produce the greatest revenue?

170

Step-by-step explanation:

The bus company charges $120 for each passenger up to 100. They charge $0.50 less per person over 100. This means for 101 passengers they charge $119.50, for 102 passengers they charge $119 etc. The total revenue for the bus company will be the price of each ticket times the number of passengers. This is shown below in the table.

Passengers Revenue

100 100*120 = 12000

101 101*119.50 = 12069.5

102 102*119 = 12138

103 103*118.50 = 12205.5

104 104*118 = 12272

105 105*117.50 = 12337.5

...

150 150*95 = 14250

...

170 170*85 = 14450

...

175 175 * 82.5 = 14437.5

The max revenue the company receives is 170 passengers.