determine revenue function, profit function, maximum profit, number of ticket sales that yields the maximum profit

Question

A charter bus company has determined that the cost, in dollars, of providing x people a tour is C(x) = 180 + 2.50x. A full tour consists of 64 people. The ticket price per person is $16 plus $0.25 for each unsold ticket. 

Answer 1

Let's first look at the revenue function.  If one ticket sells for 16 + 0.25(64-x), then x tickets sells for the price of one ticket times x tickets, or 

 

R(x) = (16 + 0.25(64-x))x = 16x + 16x - 0.25x² = 32x - 0.25x²

 

Then the profit function is the revenue function minus the cost function, which is

 

P(x) = R(x) - C(x) = (32x - 0.25x²) - (180 + 2.5x) = -0.25x² + 29.5x - 180

 

Now that we have the profit function, we can find where the profit is maximized.  Solve for the x-intercepts using the quadratic equation.  x = (-29.5 ± √690.25) / -0.5, which is approximately 6.4547813783213925 and 111.5452186216786.

 

Now we know the x-intercepts, we can find the number of ticket sales that yields the maximum profit.  The number of ticket sales that will maximize profit is the midpoint between the two x-intercepts.  When you find the midpoint, you find that the max profit is achieved when 59 tickets are sold.  If less or more are sold, you see that the profit will be less.