Find the maximum area of the largest rectangle that can be insricbed in the region bound by the graph of y=(7-x)/(6+x) and the coordinate axes

To find the maximum area of the largest rectangle that can be inscribed in the region bounded by the graph of \( y = \frac{7-x}{6+x} \) and the coordinate axes, let's proceed step by step.

### 1. Identifying the Rectangle's Dimensions
The rectangle is inscribed in the region, so its sides are parallel to the coordinate axes. Suppose the rectangle has its lower left corner at the origin \((0, 0)\) and its upper right corner at \((x, y)\), where \(y\) is determined by the function \( y = \frac{7-x}{6+x} \).

Thus, the area \(A\) of the rectangle can be expressed as:
\[
A = x \cdot y = x \cdot \frac{7-x}{6+x}
\]

### 2. Expressing the Area Function
Substitute the expression for \(y\) into the area function:
\[
A(x) = x \cdot \frac{7-x}{6+x} = \frac{x(7-x)}{6+x}
\]

Expanding and simplifying:
\[
A(x) = \frac{7x - x^2}{6+x}
\]

### 3. Maximizing the Area Function
To find the maximum area, we need to find the critical points of \( A(x) \). This requires taking the derivative of \( A(x) \) with respect to \(x\) and setting it equal to zero.

First, apply the quotient rule:
\[
A'(x) = \frac{(6+x)(7-2x) - (7x-x^2)(1)}{(6+x)^2}
\]

Simplifying the numerator:
\[
A'(x) = \frac{(42 - 12x + 7x - 2x^2) - (7x - x^2)}{(6+x)^2} = \frac{42 - 5x - x^2}{(6+x)^2}
\]

Set \( A'(x) = 0 \) to find the critical points:
\[
42 - 5x - x^2 = 0
\]

This is a quadratic equation:
\[
x^2 + 5x - 42 = 0
\]

Solve for \(x\) using the quadratic formula:
\[
x = \frac{-5 \pm \sqrt{5^2 + 4 \cdot 1 \cdot 42}}{2 \cdot 1} = \frac{-5 \pm \sqrt{25 + 168}}{2} = \frac{-5 \pm \sqrt{193}}{2}
\]

The solutions are:
\[
x = \frac{-5 + \sqrt{193}}{2} \quad \text{and} \quad x = \frac{-5 - \sqrt{193}}{2}
\]

Since \(x\) must be positive, we take:
\[
x = \frac{-5 + \sqrt{193}}{2}
\]

### 4. Calculate the Corresponding Area
Substitute this value of \(x\) back into the area function to find the maximum area. Although the exact area can be evaluated numerically, we'll keep it in terms of \( \sqrt{193} \) for now.

\[
A_{\text{max}} = \frac{\left(\frac{-5 + \sqrt{193}}{2}\right)\left(7 - \frac{-5 + \sqrt{193}}{2}\right)}{6 + \frac{-5 + \sqrt{193}}{2}}
\]

### 5. Conclusion
The maximum area of the rectangle is given by the above expression, which can be simplified further or calculated numerically for a specific value.

Would you like more details on solving the quadratic equation, simplifying the maximum area expression, or any other steps? 

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Here are 5 related questions you might consider:
1. What is the significance of the derivative in finding maximum or minimum values?
2. How does the constraint on \(x\) influence the final solution?
3. How would the problem change if the function defining the boundary were different?
4. What methods can be used to approximate the value of \( \sqrt{193} \)?
5. How does the shape of the function \( y = \frac{7-x}{6+x} \) influence the shape of the inscribed rectangle?

**Tip:** Always check the domain of your function when solving for maxima or minima, especially in real-world applications like this.

Discover how to maximize the area of a rectangle inscribed in the region bounded by the graph y = (7-x)/(6+x) and the coordinate axes. Learn step-by-step with detailed formulas and solutions. Suitable for advanced high school students and beyond.

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