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72. Break-even analysis. Use the revenue function from Problem 70, and the given cost function: R(x) = x(2,000 — 60x) - C(x) = 4,000+ 500x Revenue function Cost function where x is thousands of computers, and C(x) and R(x) are in thousands of dollars. Both functions have domain 1 ≤ x ≤ 25. (A) Sketch a graph of both functions in the same rectangular coordinate system. (B) Find the break-even points. (C) For what values of x will a loss occur? A profit?

72. Break-even analysis. Use the revenue function from Problem 70, and the given cost function: R(x) = x(2,000 — 60x) - C(x) = 4,000+ 500x Revenue function Cost function where x is thousands of computers, and C(x) and R(x) are in thousands of dollars. Both functions have domain 1 ≤ x ≤ 25. (A) Sketch a graph of both functions in the same rectangular coordinate system. (B) Find the break-even points. (C) For what values of x will a loss occur? A profit?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcription for Educational Website: --- **Revenue Analysis and Graphing** **Problem 70: Revenue** The marketing research department for a company that manufactures and sells notebook computers has established the following price–demand and revenue functions: - **Price–Demand Function:** \( p(x) = 2,000 - 60x \) - **Revenue Function:** \[ R(x) = xp(x) = x(2,000 - 60x) \] Where \( p(x) \) is the wholesale price in dollars at which \( x \) thousand computers can be sold, and \( R(x) \) is in thousands of dollars. Both functions have the domain \( 1 \leq x \leq 25 \). **Tasks:** (A) **Graph the Revenue Function:** Sketch a graph of the revenue function in a rectangular coordinate system. (B) **Maximize Revenue:** Find the value of \( x \) that will produce the maximum revenue. Determine what the maximum revenue is, rounded to the nearest thousand dollars. (C) **Optimal Pricing:** Identify the wholesale price per computer (rounded to the nearest dollar) that produces the maximum revenue. --- **Note:** The task requires understanding and manipulating quadratic functions, as well as interpreting the results in the context of business strategies for optimizing revenue and pricing.
Transcribed Image Text:Transcription for Educational Website: --- **Revenue Analysis and Graphing** **Problem 70: Revenue** The marketing research department for a company that manufactures and sells notebook computers has established the following price–demand and revenue functions: - **Price–Demand Function:** \( p(x) = 2,000 - 60x \) - **Revenue Function:** \[ R(x) = xp(x) = x(2,000 - 60x) \] Where \( p(x) \) is the wholesale price in dollars at which \( x \) thousand computers can be sold, and \( R(x) \) is in thousands of dollars. Both functions have the domain \( 1 \leq x \leq 25 \). **Tasks:** (A) **Graph the Revenue Function:** Sketch a graph of the revenue function in a rectangular coordinate system. (B) **Maximize Revenue:** Find the value of \( x \) that will produce the maximum revenue. Determine what the maximum revenue is, rounded to the nearest thousand dollars. (C) **Optimal Pricing:** Identify the wholesale price per computer (rounded to the nearest dollar) that produces the maximum revenue. --- **Note:** The task requires understanding and manipulating quadratic functions, as well as interpreting the results in the context of business strategies for optimizing revenue and pricing.
Sure, here’s the transcription for an educational website: --- ### Break-even and Profit-Loss Analysis **Problem 72: Break-even analysis.** Use the revenue function from Problem 70, and the given cost function: - **Revenue function:** \( R(x) = x(2,000 - 60x) \) - **Cost function:** \( C(x) = 4,000 + 500x \) where \( x \) is thousands of computers, and \( C(x) \) and \( R(x) \) are in thousands of dollars. Both functions have domain \( 1 \leq x \leq 25 \). Tasks: - (A) Sketch a graph of both functions in the same rectangular coordinate system. - (B) Find the break-even points. - (C) For what values of \( x \) will a loss occur? A profit? --- **Problem 73: Profit-loss analysis.** Use the revenue and cost functions from Problem 71: - **Revenue function:** \( R(x) = x(75 - 3x) \) - **Cost function:** \( C(x) = 125 + 16x \) where \( x \) is in millions of chips, and \( R(x) \) and \( C(x) \) are in millions of dollars. Both functions have domain \( 1 \leq x \leq 20 \). Tasks: - (A) Sketch a graph of both functions in the same rectangular coordinate system. - (B) Find the break-even points to the nearest thousand chips. - (C) For what values of \( x \) will a loss occur? A profit? --- **Explanation of Graphs/Diagrams:** For each problem, you are expected to draw two graphs on the same coordinate plane. Both graphs represent linear equations derived from the revenue and cost functions. The intersection points of these two graphs indicate the break-even points, where revenue equals cost. Profits occur when the revenue graph lies above the cost graph, while losses occur when the cost graph lies above the revenue graph. ---
Transcribed Image Text:Sure, here’s the transcription for an educational website: --- ### Break-even and Profit-Loss Analysis **Problem 72: Break-even analysis.** Use the revenue function from Problem 70, and the given cost function: - **Revenue function:** \( R(x) = x(2,000 - 60x) \) - **Cost function:** \( C(x) = 4,000 + 500x \) where \( x \) is thousands of computers, and \( C(x) \) and \( R(x) \) are in thousands of dollars. Both functions have domain \( 1 \leq x \leq 25 \). Tasks: - (A) Sketch a graph of both functions in the same rectangular coordinate system. - (B) Find the break-even points. - (C) For what values of \( x \) will a loss occur? A profit? --- **Problem 73: Profit-loss analysis.** Use the revenue and cost functions from Problem 71: - **Revenue function:** \( R(x) = x(75 - 3x) \) - **Cost function:** \( C(x) = 125 + 16x \) where \( x \) is in millions of chips, and \( R(x) \) and \( C(x) \) are in millions of dollars. Both functions have domain \( 1 \leq x \leq 20 \). Tasks: - (A) Sketch a graph of both functions in the same rectangular coordinate system. - (B) Find the break-even points to the nearest thousand chips. - (C) For what values of \( x \) will a loss occur? A profit? --- **Explanation of Graphs/Diagrams:** For each problem, you are expected to draw two graphs on the same coordinate plane. Both graphs represent linear equations derived from the revenue and cost functions. The intersection points of these two graphs indicate the break-even points, where revenue equals cost. Profits occur when the revenue graph lies above the cost graph, while losses occur when the cost graph lies above the revenue graph. ---
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