Problem

A concert promoter sells tickets and has a marginal-profit function given below, where $\mathrm{P}^{\prime}(\mathrm{x})$ 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 $P^{\prime}(x)$. Find the total profit from the sale of the first 90 tickets, disregarding any fixed costs. \[ P^{\prime}(x)=9 x-1084 \] The total profit is $\$$ (Round to the nearest cent as needed.)

Solution

Step 1 :Given the marginal-profit function \(P^{\prime}(x)=9x-1084\), where \(P^{\prime}(x)\) is in dollars per ticket and \(x\) is the number of tickets sold.

Step 2 :The total profit from the sale of the first 90 tickets is the integral of the marginal profit function from 0 to 90.

Step 3 :Calculate the integral of \(P^{\prime}(x)=9x-1084\) from 0 to 90 to find the total profit.

Step 4 :The total profit from the sale of the first 90 tickets is -61110 dollars.

Step 5 :However, it's unusual for a profit to be negative. This could mean that the costs associated with selling the tickets (such as production costs, marketing, etc.) are greater than the revenue from ticket sales.

Step 6 :Final Answer: The total profit from the sale of the first 90 tickets is \(\boxed{-\$61110}\).

From Solvely APP
Source: https://solvelyapp.com/problems/7940/

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