Problem 39 - Algebra A Combined Function

Question

The total revenue from the sale of a popular book is approximated by the rational function \(R(x)=\frac{1000 x^{2}}{x^{2}+4},\) where \(x\) is the number of years since publication and \(R(x)\) is the total revenue in millions of dollars. a. Find the total revenue at the end of the first year. b. Find the total revenue at the end of the second year. c. Find the revenue during the second year only.

Step-by-Step Solution

Answer

a. $200$ million, b. $500$ million, c. $300$ million.

Step 1: Understand the Function

The given function is a rational function \( R(x) = \frac{1000x^2}{x^2 + 4} \). Here, \( x \) is the number of years since publication and \( R(x) \) is the total revenue in millions of dollars.

Step 2: Calculate Revenue at End of First Year

To find the revenue at the end of the first year, substitute \( x = 1 \) into the function:\[ R(1) = \frac{1000(1)^2}{(1)^2 + 4} = \frac{1000 \, \cdot \, 1}{1 + 4} = \frac{1000}{5} = 200 \text{ million dollars} \].

Step 3: Calculate Revenue at End of Second Year

To find the revenue at the end of the second year, substitute \( x = 2 \) into the function:\[ R(2) = \frac{1000(2)^2}{(2)^2 + 4} = \frac{1000 \, \cdot \, 4}{4 + 4} = \frac{4000}{8} = 500 \text{ million dollars} \].

Step 4: Calculate Revenue During Second Year

The revenue during the second year is the difference between the total revenue at the end of the second year and the total revenue at the end of the first year:\[ R(2) - R(1) = 500 - 200 = 300 \text{ million dollars} \].

Key Concepts

Polynomial Expressions

Polynomial expressions form the backbone of many mathematical functions, including rational functions. In our exercise, we see a polynomial expression in both the numerator and the denominator of the rational function, \( R(x) = \frac{1000x^2}{x^2 + 4} \). Polynomials like \( 1000x^2 \) and \( x^2 + 4 \) consist of variables, coefficients, and exponents. They can be manipulated through addition, subtraction, multiplication, and division.
To understand rational functions, which are quotients of two polynomial expressions, it's essential to identify the parts. The numerator, \( 1000x^2 \), is straightforward: it grows rapidly as \( x \) increases. The denominator, \( x^2 + 4 \), affects the rate of growth, ensuring the overall function remains bounded as \( x \) gets large.
Understanding polynomial expressions is critical for solving and simplifying rational functions. They allow us to grasp how changes in \( x \) influence the function's outcome, which in this case, affects revenue prediction over time.

Function Evaluation

In mathematics, function evaluation involves finding the output of a function for a specific input value. For our rational function \( R(x) = \frac{1000x^2}{x^2 + 4} \), this means substituting a value for \( x \) and simplifying the expression to find \( R(x) \).
For example, to find the revenue at the end of the first year, we substitute \( x = 1 \) into the function:

• Calculate \( 1000(1)^2 \) to get 1000.
• Compute \( (1)^2 + 4 \) to get 5.
• Divide the results: \( \frac{1000}{5} = 200 \).

This gives us a revenue of 200 million dollars at the end of the first year. Replacing \( x \) with 2 allows us to find the revenue at the end of the second year. Thus, function evaluation is a critical process in understanding how inputs impact numerical predictions and interpretations.

Revenue Calculation

Revenue calculation, in this exercise, uses a rational function to estimate income over time. By evaluating \( R(x) \), we determine the total income generated by a book over a defined period.
At the end of the first year, we found the revenue to be 200 million dollars, as calculated using \( R(1) \). Moving to the second year, using \( R(2) \), we estimated the revenue to be 500 million dollars.
The task of finding the revenue for just the second year requires understanding not only separate year evaluations but also how to calculate year-over-year change:

• Revenue for the second year = \( R(2) - R(1) \), which results in 300 million dollars.

This way, revenue calculation using rational functions helps businesses, economists, and analysts gauge financial performance efficiently for strategic planning and evaluation.