A concert promoter sells tickets and has a marginal-profit function given below, where \( P^{\prime}(x) \) is in dollars per ticket. This means that the number of tickets sold, \( x \), is \( P^{\prime}(x) \). Find the total profit from the sale of the first 80 tickets, disregarding any fixed costs. \( P^{\prime}(x)=9 x-1162 \) The total profit is \( \$ \square \). (Round to the nearest cent as needed.)

To find the total profit from the sale of the first 80 tickets, we need to calculate the total profit function and then evaluate it at \( x = 80 \). Given: Marginal-profit function: \( P^{\prime}(x) = 9x - 1162 \) To find the total profit function, we need to integrate the marginal-profit function with respect to \( x \). The total profit function is given by: \[ P(x) = \int P^{\prime}(x) \, dx \] Integrating \( P^{\prime}(x) = 9x - 1162 \) with respect to \( x \) gives: \[ P(x) = \frac{9}{2}x^2 - 1162x + C \] To find the constant of integration \( C \), we can use the fact that the total profit at \( x = 0 \) is 0. This is because the total profit at the start is 0. Substitute \( x = 0 \) into the total profit function: \[ P(0) = \frac{9}{2}(0)^2 - 1162(0) + C = 0 \] \[ C = 0 \] Therefore, the total profit function is: \[ P(x) = \frac{9}{2}x^2 - 1162x \] Now, we can find the total profit from the sale of the first 80 tickets by evaluating the total profit function at \( x = 80 \): \[ P(80) = \frac{9}{2}(80)^2 - 1162(80) \] Let's calculate the total profit. Calculate the value by following steps: - step0: Calculate: \(\frac{9}{2}\times 80^{2}-1162\times 80\) - step1: Multiply the numbers: \(28800-1162\times 80\) - step2: Multiply the numbers: \(28800-92960\) - step3: Subtract the numbers: \(-64160\) The total profit from the sale of the first 80 tickets is $-64160. Therefore, the total profit is $-64160.