The marginal revenue from the sale of x cans of air freshener is given by 9.8−0.35√x
dollars per can. Find the increase in total revenue if the number of cans sold is increased from 153
to 198
. Round your answer to the nearest cent.

To find the increase in total revenue when the number of cans sold increases from 153 to 198, we need to integrate the marginal revenue function $ 9.8 - 0.35\sqrt{x} $ over the interval from 153 to 198.

The increase in total revenue $ \Delta R $ can be calculated as follows:

\[
\Delta R = \int_{153}^{198} \left( 9.8 - 0.35\sqrt{x} \right) \, dx
\]

First, let's break down the integral:

\[
\int_{153}^{198} \left( 9.8 - 0.35\sqrt{x} \right) \, dx = \int_{153}^{198} 9.8 \, dx - \int_{153}^{198} 0.35\sqrt{x} \, dx
\]

Now, we solve each integral separately.

1. The integral of a constant:

\[
\int_{153}^{198} 9.8 \, dx = 9.8 \left[ x \right]_{153}^{198} = 9.8 \left( 198 - 153 \right) = 9.8 \times 45 = 441
\]

2. The integral of $ \sqrt{x} $:

\[
\int_{153}^{198} \sqrt{x} \, dx = \int_{153}^{198} x^{1/2} \, dx
\]

Using the power rule of integration:

\[
\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C
\]

So,

\[
\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}
\]

Evaluating this from 153 to 198:

\[
\left[ \frac{2}{3} x^{3/2} \right]_{153}^{198} = \frac{2}{3} \left( 198^{3/2} - 153^{3/2} \right)
\]

We need to calculate $ 198^{3/2} $ and $ 153^{3/2} $:

\[
198^{3/2} = (198^{1/2})^3 = (\sqrt{198})^3 \approx (14.071)^3 \approx 2783.905
\]

\[
153^{3/2} = (153^{1/2})^3 = (\sqrt{153})^3 \approx (12.369)^3 \approx 1893.896
\]

So,

\[
\frac{2}{3} \left( 198^{3/2} - 153^{3/2} \right) = \frac{2}{3} \left( 2783.905 - 1893.896 \right) = \frac{2}{3} \times 890.009 \approx 593.34
\]

Now, multiplying by the constant:

\[
0.35 \times 593.34 \approx 207.67
\]

Finally, we find the increase in total revenue:

\[
\Delta R = 441 - 207.67 = 233.33
\]

Thus, the increase in total revenue if the number of cans sold is increased from 153 to 198 is approximately $ \$233.33 $.

Do you want more details or have any questions? Here are 5 more questions you might find interesting:

1. How do you find the derivative of a polynomial function?
2. What are the steps to solve a system of linear equations using matrix methods?
3. How do you apply the chain rule in calculus?
4. What is the integral of a trigonometric function like $\sin(x)$ or $\cos(x)$?
5. How do you determine the convergence of an infinite series?

Tip: Always remember to check the domain of the function before integrating to ensure the limits of integration are within the valid range of the function.

Learn how to calculate the increase in total revenue using the marginal revenue function. This problem involves integrating a marginal revenue function to find the change in revenue when the number of units sold increases. Suitable for advanced high school students and beyond.

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