Business Calculus Final Exam Study Guide – Step-by-Step Guidance

Q2. A company manufactures memory chips for microcomputers. Its marketing research department, using statistical techniques, collected the data shown in the table below, where $p$ is the wholesale price per chip at which $x$ million chips can be sold. Using special analytical techniques, an analyst produced the following price-demand function to model the data: $p(x) = 75 - 3x$, where $1 \leq x \leq 20$.

Background

Topic: Linear Functions and Revenue Modeling

This question tests your ability to interpret and graph linear price-demand functions, estimate prices for given demand levels, and construct and analyze revenue functions in a business context.

Key Terms and Formulas

• Price-demand function: $p(x) = 75 - 3x$

• Revenue function: $R(x) = p(x) \cdot x$

• Domain: $1 \leq x \leq 20$ (million chips)

Step-by-Step Guidance

1. Understand that the price-demand function $p(x) = 75 - 3x$ is linear, meaning price decreases as demand increases.

2. To estimate the price for a specific demand, substitute the value of $x$ (e.g., $x = 7$ or $x = 11$) into the price-demand function.

3. To write the revenue function, multiply the price-demand function by the number of units sold: $R(x) = (75 - 3x)x$.

4. To graph the price-demand function, plot points for $x$ values within the domain and connect them to show the linear relationship. For the revenue function, plot $R(x)$ for $x$ values in the domain to visualize how revenue changes with demand.

Try solving on your own before revealing the answer!

Final Answer:

(b) For $x = 7$, $p(7) = 75 - 3 \times 7 = 54$; for $x = 11$, $p(11) = 75 - 3 \times 11 = 42$.

(c) Revenue function: $R(x) = 75x - 3x^2$.

The price-demand graph is linear and the revenue graph is a downward-opening parabola, showing maximum revenue at a certain demand level.