Question

The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function $$ T(x)=\frac{120 x^{2}}{x^{2}+4} $$ where $T(x)$ is measured in millions of dollars and $x$ is the number of months since the movie's release. a. What are the total box-office receipts after the first month? The second month? The third month? b. What will the movie gross in the long run (when $x$ is very large)?

Name: Problem 85, Chapter 9: Applied Mathematics: For the Managerial, Life, and Social Sciences
Uploaded: 2021-01-22T14:07:24-08:00
Duration: 3 min 8 s
Channel: Adam Dehollander

   The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function
$$
T(x)=\frac{120 x^{2}}{x^{2}+4}
$$
where $T(x)$ is measured in millions of dollars and $x$ is the number of months since the movie's release.
a. What are the total box-office receipts after the first month? The second month? The third month?
b. What will the movie gross in the long run (when $x$ is very large)?

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan 5th Edition

Chapter 9, Problem 85

Instant Answer

Solved by Expert Adam Dehollander

01/22/2021

Step 1

To find $T(1)$, substitute $x = 1$ into the function: $$ T(1) = \frac{120 \times 1^2}{1^2 + 4} = \frac{120 \times 1}{1 + 4} = \frac{120}{5} = 24. $$ Thus, the total box-office receipts after the first month are $24$ million dollars. Show more…

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The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function $$ T(x)=\frac{120 x^{2}}{x^{2}+4} $$ where $T(x)$ is measured in millions of dollars and $x$ is the number of months since the movie's release. a. What are the total box-office receipts after the first month? The second month? The third month? b. What will the movie gross in the long run (when $x$ is very large)?

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Key Concepts

Function Evaluation

This concept involves substituting particular values into a function to determine its output at those points. In this context, you evaluate the function at specific months to calculate the box-office receipts for each month after the movie's release.

Rational Functions

Rational functions are ratios of polynomial functions. They are characterized by their asymptotic behavior, potential discontinuities, and the relative degrees of the numerator and denominator, which help determine properties such as growth rates and limits.

Limits and Asymptotic Behavior

This concept entails determining the behavior of a function as the input approaches a particular value, often infinity. For rational functions, examining limits can reveal horizontal asymptotes, indicating the long-run behavior or the value that the function approaches as the independent variable becomes very large.

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