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The problem you uploaded asks to maximize the total revenue \( R(x, y) \), where the revenue function is given as:

\[
R(x, y) = -4x^2 - 8y^2 - 2xy + 44x + 104y
\]

where \( x \) represents the number of jogging shoes (in thousands) and \( y \) represents the number of cross trainers (in thousands).

### Steps to Maximize the Revenue:

1. **Find the partial derivatives of \( R(x, y) \) with respect to \( x \) and \( y \)**:
   - \( R_x = \frac{\partial R}{\partial x} \)
   - \( R_y = \frac{\partial R}{\partial y} \)

2. **Set the partial derivatives equal to zero** to find the critical points:
   - Solve \( R_x = 0 \)
   - Solve \( R_y = 0 \)

3. **Solve the system of equations** from the partial derivatives to find the values of \( x \) and \( y \) that maximize the revenue.

Let me compute the partial derivatives and solve for the critical points.The values of \( x \) and \( y \) that maximize the total revenue are:

\[
(x, y) = (4, 6)
\]

This means the company should produce 4,000 units of jogging shoes and 6,000 units of cross trainers to maximize their revenue.

Would you like more details on the process or have any questions?

Here are 5 related questions you might find interesting:

1. How does the second derivative test confirm that this is a maximum?
2. What happens to revenue if the company produces fewer than 4,000 jogging shoes?
3. What is the total maximum revenue achieved at these values?
4. Can we solve this problem using a different method, such as Lagrange multipliers?
5. What if the company wanted to minimize the cost instead of maximizing revenue?

**Tip**: When solving optimization problems, always check whether the critical points yield a maximum or minimum by using second derivative tests or boundary analysis.

A company manufactures two types of athletic shoes, jogging shoes, and cross trainers. The total revenue from x units of jogging shoes and y units of cross trainers is given by the equation: R(x, y) = -4x^2 - 8y^2 - 2xy + 44x + 104y. Find the values of x and y to maximize the total revenue.

This problem involves maximizing the total revenue from jogging shoes and cross trainers using a quadratic revenue function. Learn how to use partial derivatives and optimization techniques to solve this business-related problem, suitable for high school students.

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