Optimization Problem: A rectangle is inscribed under the parabola y

To solve the optimization problem of finding the dimensions of a rectangle inscribed under the parabola $y = 4 - x^2$, we maximize its area.

Problem Setup:

1. Parabola Equation: $y = 4 - x^2$.

2. The rectangle is symmetric about the y-axis, so consider only the positive x-axis for simplification.

3. The rectangle's dimensions are determined by:

   • Width: $2x$ (from $-x$ to $x$).
   • Height: $y = 4 - x^2$ (determined by the parabola at $x$).

4. Area Function: The area $A$ of the rectangle is: $A = \text{width} \times \text{height} = (2x)(4 - x^2).$ Simplifying: $A(x) = 8x - 2x^3.$

Step 1: Find the Critical Points

To maximize the area, differentiate $A(x)$ with respect to $x$: $A'(x) = 8 - 6x^2.$ Set $A'(x) = 0$ to find critical points: $8 - 6x^2 = 0 \implies 6x^2 = 8 \implies x^2 = \frac{4}{3} \implies x = \pm \sqrt{\frac{4}{3}}.$ Since $x > 0$, we take: $x = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}.$

Step 2: Verify Maximum Using Second Derivative

The second derivative of $A(x)$ is: $A''(x) = -12x.$ At $x = \frac{2}{\sqrt{3}} > 0$, $A''(x) < 0$, confirming a local maximum.

Step 3: Dimensions of the Rectangle

1. Width: $2x = 2 \times \frac{2}{\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}$ (after rationalizing).
2. Height: $y = 4 - x^2 = 4 - \frac{4}{3} = \frac{12}{3} - \frac{4}{3} = \frac{8}{3}$.

Final Answer:

The dimensions of the rectangle that maximize the area are:

• Width: $\frac{4\sqrt{3}}{3}$,
• Height: $\frac{8}{3}$.

The maximum area is: $A_{\text{max}} = (2x)(y) = \left( \frac{4\sqrt{3}}{3} \right) \left( \frac{8}{3} \right) = \frac{32\sqrt{3}}{9}.$

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Related Questions:

1. How do we interpret critical points geometrically in optimization problems?
2. Can the second derivative test fail in some cases?
3. How would the solution change if the rectangle wasn’t symmetric about the y-axis?
4. What is the relationship between symmetry and optimization in this problem?
5. How would adding a constraint to the parabola change the solution?

Tip:

Always check the endpoints and behavior of the function over its domain when solving optimization problems!