A retail company estimates that if it spends x thousands of dollars on advertising during the year, it will realize a profit of P(x) dollars, where P(æ) - 0.25x + 80x + 1100, where 0 < x < 210. a. What is the company's marginal profit at the $120000 and $190000 advertising levels? P'(120) P'(190) = b. What advertising expenditure would you recommend to this company?
A retail company estimates that if it spends x thousands of dollars on advertising during the year, it will realize a profit of P(x) dollars, where P(æ) - 0.25x + 80x + 1100, where 0 < x < 210. a. What is the company's marginal profit at the $120000 and $190000 advertising levels? P'(120) P'(190) = b. What advertising expenditure would you recommend to this company?
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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Transcribed Image Text:**Problem Statement:**
A retail company estimates that if it spends \( x \) thousands of dollars on advertising during the year, it will realize a profit of \( P(x) \) dollars, where \( P(x) = -0.25x^2 + 80x + 1100 \), where \( 0 \leq x \leq 210 \).
**Questions:**
a. What is the company's marginal profit at the $120,000 and $190,000 advertising levels?
\( P'(120) = \) \_\_\_\_\_
\( P'(190) = \) \_\_\_\_\_
b. What advertising expenditure would you recommend to this company?
\$ \_\_\_\_\_
**Explanations:**
- The company's profit \( P(x) \) is modeled as a quadratic function of the advertising budget \( x \).
- The marginal profit is found by differentiating the profit function \( P(x) \) with respect to \( x \) and evaluating it at specific advertising levels \( x \).
- \( P'(x) \) gives the rate of change of profit with respect to changes in the advertising budget.
- Solving question (b) involves determining the advertising expenditure that maximizes profit, which often requires finding the vertex of the quadratic function.
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