뒤로Business Calculus: Maximizing Revenue with Price-Demand Functions
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Q19. A company manufactures and sells x smartphones per week. The weekly price-demand and cost equations are, respectively, and .
A1) What price should the company charge for phones,
A2) and how many phones should be produced to maximize the weekly revenue?
A3) What is the maximum weekly revenue?
Background
Topic: Revenue Maximization using Price-Demand Functions (Business Calculus)
This question tests your ability to use price-demand equations to find the price and quantity that maximize revenue, a classic application of calculus in business. You will also use cost functions, but the focus here is on maximizing revenue, not profit.
Key Terms and Formulas
Price-demand equation: Relates the price to the number of units sold.
Revenue function:
To maximize revenue: Find that maximizes (usually by finding where ).
Step-by-Step Guidance
Write the revenue function in terms of using the price-demand equation: .
Substitute the given price-demand equation into the revenue function to get as a function of only.
Expand and simplify to get a quadratic function.
To find the value of that maximizes revenue, take the derivative and set it equal to zero.
Solve for to find the number of phones that should be produced and sold to maximize revenue. Then, use the price-demand equation to find the corresponding price.
To find the maximum revenue, substitute the value of back into the revenue function .
Try solving on your own before revealing the answer!
Final Answers:
A1) Price to charge:
A2) Number of phones to produce and sell:
A3) Maximum weekly revenue:
We set up the revenue function , found its maximum by taking the derivative and setting it to zero, and then evaluated the price and revenue at that value.
Q20. A company manufactures and sells x digital cameras per week. The weekly price-demand and cost equations are, respectively, and .
A1) What price should the company charge for cameras,
A2) and how many cameras should be produced to maximize the weekly revenue?
A3) What is the maximum weekly revenue?
Background
Topic: Revenue Maximization using Price-Demand Functions (Business Calculus)
This question is similar to the previous one, focusing on maximizing revenue using a price-demand equation. The process involves forming the revenue function, finding its maximum, and interpreting the results in a business context.
Key Terms and Formulas
Price-demand equation:
Revenue function:
To maximize revenue: Find that maximizes (set ).
Step-by-Step Guidance
Write the revenue function: .
Substitute the price-demand equation into the revenue function to get in terms of .
Expand and simplify to a quadratic function.
Take the derivative and set it equal to zero to find the critical point(s).
Solve for to find the number of cameras to produce and sell for maximum revenue. Then, use the price-demand equation to find the corresponding price.
Substitute this value into to find the maximum weekly revenue.
Try solving on your own before revealing the answer!
Final Answers:
A1) Price to charge:
A2) Number of cameras to produce and sell:
A3) Maximum weekly revenue:
We set up the revenue function , found its maximum by taking the derivative and setting it to zero, and then evaluated the price and revenue at that value.
