Math / Calculus

QuestionA concert promoter sells tickets and has a marginal-profit function given below, where $P^{\prime}(x)$ is in dollars 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 80 tickets, disregarding any fixed costs. $P^{\prime}(x)=9 x-1129$
The total profit is $\$$ $\square$ (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

1. The marginal-profit function is given by $P^{\prime}(x) = 9x - 1129$ .
2. We need to find the total profit from selling the first 80 tickets.
3. Fixed costs are disregarded.

STEP 2

1. Understand the relationship between marginal profit and total profit.
2. Set up the integral to find total profit.
3. Evaluate the integral.
4. Calculate the total profit for the first 80 tickets.

STEP 3

Understand the relationship between marginal profit and total profit:
The total profit $P(x)$ is the integral of the marginal-profit function $P^{\prime}(x)$ . Therefore, to find the total profit from selling the first 80 tickets, we need to integrate $P^{\prime}(x)$ from 0 to 80.

STEP 4

Set up the integral to find total profit:
$P(x) = \int P^{\prime}(x) \, dx = \int (9x - 1129) \, dx$
We will evaluate this integral from 0 to 80.

STEP 5

Evaluate the integral:
$P(x) = \int (9x - 1129) \, dx = \left[ \frac{9}{2}x^2 - 1129x \right]$
Evaluate from 0 to 80:
$P(80) = \left[ \frac{9}{2}(80)^2 - 1129(80) \right] - \left[ \frac{9}{2}(0)^2 - 1129(0) \right]$

STEP 6

Calculate the total profit for the first 80 tickets:
$P(80) = \left[ \frac{9}{2}(6400) - 1129(80) \right]$ $= \left[ 28800 - 90320 \right]$ $= -61520$
The total profit from the sale of the first 80 tickets is:
$\boxed{-61520}$
Since the result is negative, it indicates a loss rather than a profit.

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Studdy Solution

STEP 1

1. The marginal-profit function is given by $P^{\prime}(x) = 9x - 1129$ .
2. We need to find the total profit from selling the first 80 tickets.
3. Fixed costs are disregarded.

STEP 2

1. Understand the relationship between marginal profit and total profit.
2. Set up the integral to find total profit.
3. Evaluate the integral.
4. Calculate the total profit for the first 80 tickets.

STEP 3

Understand the relationship between marginal profit and total profit:
The total profit $P(x)$ is the integral of the marginal-profit function $P^{\prime}(x)$ . Therefore, to find the total profit from selling the first 80 tickets, we need to integrate $P^{\prime}(x)$ from 0 to 80.

STEP 4

Set up the integral to find total profit:
$P(x) = \int P^{\prime}(x) \, dx = \int (9x - 1129) \, dx$
We will evaluate this integral from 0 to 80.

STEP 5

Evaluate the integral:
$P(x) = \int (9x - 1129) \, dx = \left[ \frac{9}{2}x^2 - 1129x \right]$
Evaluate from 0 to 80:
$P(80) = \left[ \frac{9}{2}(80)^2 - 1129(80) \right] - \left[ \frac{9}{2}(0)^2 - 1129(0) \right]$

STEP 6

Calculate the total profit for the first 80 tickets:
$P(80) = \left[ \frac{9}{2}(6400) - 1129(80) \right]$ $= \left[ 28800 - 90320 \right]$ $= -61520$
The total profit from the sale of the first 80 tickets is:
$\boxed{-61520}$
Since the result is negative, it indicates a loss rather than a profit.

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Studdy Solution

STEP 1

1. The marginal-profit function is given by P′(x)=9x−1129 P^{\prime}(x) = 9x - 1129 P′(x)=9x−1129.2. We need to find the total profit from selling the first 80 tickets.3. Fixed costs are disregarded.

STEP 2

1. Understand the relationship between marginal profit and total profit.2. Set up the integral to find total profit.3. Evaluate the integral.4. Calculate the total profit for the first 80 tickets.

STEP 3

STEP 4

Set up the integral to find total profit:P(x)=∫P′(x) dx=∫(9x−1129) dx P(x) = \int P^{\prime}(x) \, dx = \int (9x - 1129) \, dx P(x)=∫P′(x)dx=∫(9x−1129)dxWe will evaluate this integral from 0 to 80.

STEP 5

STEP 6

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1. The marginal-profit function is given by $P^{\prime}(x) = 9x - 1129$ .
2. We need to find the total profit from selling the first 80 tickets.
3. Fixed costs are disregarded.

1. Understand the relationship between marginal profit and total profit.
2. Set up the integral to find total profit.
3. Evaluate the integral.
4. Calculate the total profit for the first 80 tickets.

Set up the integral to find total profit:
$P(x) = \int P^{\prime}(x) \, dx = \int (9x - 1129) \, dx$
We will evaluate this integral from 0 to 80.