Question
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The annual total revenue for a product is given by dollars, where is the number of units sold. To maximize revenue, how many units must be sold? What is the
maximum possible annual revenue?
To maximize revenue, units must be sold.
(Simplify your answer.)

Ask by Wyatt Christensen. in the United States
Jan 24,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

To maximize revenue, 4,500 units must be sold, and the maximum possible annual revenue is $101,250,000.

Solution

To maximize revenue, we need to find the number of units that maximizes the revenue function:
This is a quadratic function of the form , where:
  • (since the coefficient of is negative, the parabola opens downward, indicating a maximum point)
The number of units that maximizes the revenue can be found using the vertex formula for a parabola:
Plugging in the values:
So, 4,500 units must be sold to maximize revenue.
Next, to find the maximum possible annual revenue, substitute back into the revenue function:
Calculate each term:
Subtract the second term from the first:
Summary:
  • To maximize revenue, 4,500 units must be sold.
  • The maximum possible annual revenue is $101,250,000.

Answered by UpStudy AI and reviewed by a Professional Tutor

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Simplify this solution

Beyond the Answer

To maximize revenue, we need to find the value of that maximizes the function . This is a quadratic function and opens downwards (since the coefficient of is negative), so its maximum occurs at the vertex. The vertex can be found using the formula , where and .
Plugging in the values, we get:
To find the maximum possible annual revenue, substitute back into the revenue function:
Calculating this gives:
Therefore, to maximize revenue, 4,500 units must be sold, leading to a maximum revenue of $101,250,000.

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