Over the next 15 years, a company projects its continuous flow of revenue to be $R(x)=100 e^{0.1 x}$ thousands of dollars and its costs to be $C(x)=0.9 x^{2}+90$ thousands of dollars, where $x$ represents the number of years from now. Approximate, to the nearest dollar, the profit this company can expect to make over the next 15 years. \[ P= \]
Solution
Step 1 ：Given the continuous flow of revenue to be $R(x)=100 e^{0.1 x}$ thousands of dollars and costs to be $C(x)=0.9 x^{2}+90$ thousands of dollars, where $x$ represents the number of years from now.
Step 2 ：The profit of a company is calculated by subtracting the costs from the revenue. In this case, we need to integrate the profit function over the next 15 years to get the total profit. The profit function is given by $P(x) = R(x) - C(x)$.
Step 3 ：We can substitute the given functions into this equation and then integrate from 0 to 15. So, $P(x) = -0.9x^{2} + 100e^{0.1x} - 90$.
Step 4 ：By integrating this function from 0 to 15, we get the total profit to be approximately 1119 thousand dollars.
Step 5 ：Final Answer: The company can expect to make a profit of approximately \$\boxed{1119}$ thousand over the next 15 years.
