Question

A concert promoter sells tickets and has a marginal-profit function given below, where P′(x) is in dollars per ticket. This means that the rate of change of total profit with respect to the number of tickets sold, x, is P′(x). Find the total profit from the sale of the first 190 tickets, disregarding any fixed costs. P′(x)=3x−1159 The total profit is $ (Round to the nearest cent as needed.)

Answer 1

-$165,760

Step-by-step explanation:

To find the total profit from the sale of the first 190 tickets, we need to integrate the marginal-profit function, P'(x), with respect to x from 0 to 190.

The integral of P'(x) with respect to x will give us the total profit function, P(x), and evaluating it from 0 to 190 will give us the total profit from the sale of the first 190 tickets.

Integrating P'(x) = 3x - 1159 with respect to x:

P(x) = ∫(3x - 1159) dx

= (3/2)x^2 - 1159x + C

Now, to find the total profit from the sale of the first 190 tickets, we evaluate P(x) from 0 to 190:

Profit = P(190) - P(0)

= [(3/2)(190)^2 - 1159(190)] - [(3/2)(0)^2 - 1159(0)]

= [(3/2)(36100) - 219910] - [0 - 0]

= 54150 - 219910

= -$165,760

Rounding to the nearest cent, the total profit from the sale of the first 190 tickets is -$165,760.