Here’s how you can approach solving these problems:

Problem 5: Largest Rectangle in an Ellipse

We are tasked with finding the area of the largest rectangle that can be inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

Solution Outline:

1. The vertices of the rectangle will lie on the ellipse.
2. Use the parametric representation of the ellipse: $x = a \cos \theta$, $y = b \sin \theta$.
3. The area of the rectangle is $A = 4 \cdot x \cdot y = 4ab \cos \theta \sin \theta = 2ab \sin(2\theta)$.
4. Maximize $A = 2ab \sin(2\theta)$, which is maximized when $\sin(2\theta) = 1$.
5. Thus, the maximum area is $A = 2ab$.

Problem 6: Largest Rectangle in a Right Triangle

Given a right triangle with legs of $4 \, \text{cm}$ and $6 \, \text{cm}$, find the area of the largest rectangle that can be inscribed.

Solution Outline:

1. The rectangle is inscribed such that two sides lie along the legs of the triangle.
2. If the rectangle's dimensions are $x$ and $y$, the hypotenuse constraint gives $\frac{x}{4} + \frac{y}{6} = 1$, or $y = 6 - \frac{3x}{2}$.
3. The area is $A = x \cdot y = x \left(6 - \frac{3x}{2}\right) = 6x - \frac{3x^2}{2}$.
4. Maximize $A(x)$ by setting $\frac{dA}{dx} = 0$: $\frac{dA}{dx} = 6 - 3x = 0 \implies x = 2.$
5. Substituting $x = 2$ into $y = 6 - \frac{3x}{2}$, $y = 3$.
6. The maximum area is $A = 2 \cdot 3 = 6 \, \text{cm}^2$.

Problem 7: Dividing Wire for Maximizing/Minimizing Area

A $23 \, \text{m}$ wire is divided into a square and an equilateral triangle.

Part (a) Maximize Total Area:

1. Let $x$ meters be used for the square, leaving $23 - x$ for the triangle.
2. Side of the square: $s = \frac{x}{4}$, area of the square: $A_s = \left(\frac{x}{4}\right)^2$.
3. Side of the triangle: $t = \frac{23 - x}{3}$, area of the triangle: $A_t = \frac{\sqrt{3}}{4} \left(\frac{23 - x}{3}\right)^2$.
4. Total area: $A = A_s + A_t$.
5. Differentiate $A$ with respect to $x$ and solve for $x$ to maximize $A$.

Part (b) Minimize Total Area:

• Follow the same procedure as above but solve for $x$ to minimize $A$.

Would you like me to provide detailed calculations or clarifications on any specific part?

Related Questions:

1. How do you derive the formula for the largest rectangle inscribed in an ellipse?
2. Can we generalize the method for finding the largest rectangle in any triangle?
3. How does the constraint affect the optimization in Problem 7?
4. Why does $\sin(2\theta) = 1$ maximize the area in Problem 5?
5. Could numerical methods be used to solve Problem 7 more efficiently?

Tip:

When solving optimization problems, always check the second derivative or boundaries to confirm if it’s a maximum or minimum.