Math Problem Statement

What price should the company charge to maximize revenue?

What price should the company charge to maximize revenue?

Solution

To find the price that maximizes revenue, we use the demand equation:

p=13x+100p = -\frac{1}{3}x + 100

From part (c), the quantity xx that maximizes revenue is found to be x=150x = 150.

Now, substituting x=150x = 150 into the demand equation:

p=13(150)+100p = -\frac{1}{3} (150) + 100

p=50+100p = -50 + 100

p=50p = 50

Thus, the company should charge $50 to maximize revenue.

Would you like further clarification on any part? Here are some related questions:

  1. How is the revenue function derived from the demand equation?
  2. Why does revenue maximize at x=150x = 150?
  3. What would happen to revenue if the price were set higher or lower than $50?
  4. Can we verify the maximum revenue using calculus (derivative method)?
  5. How does this type of problem relate to real-world pricing strategies?

Tip: The maximum revenue occurs at the vertex of the quadratic revenue function, which can be found using the vertex formula x=b2ax = -\frac{b}{2a}.

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Math Problem Analysis

Mathematical Concepts

Algebra
Revenue Maximization
Quadratic Functions

Formulas

Revenue function: R(x) = p * x
Demand equation: p = -1/3 x + 100
Vertex formula for a quadratic function: x = -b / 2a

Theorems

Vertex Theorem for Quadratic Functions

Suitable Grade Level

Grades 9-12