We aim to find the rectangle of largest area with its base on the $x$-axis and its top two vertices on the parabola $y = 7 - x^2$.

Step 1: Define the area of the rectangle

The rectangle is symmetric about the $y$-axis. Let the top two vertices of the rectangle be $(x, y)$ and $(-x, y)$. The height of the rectangle is $y = 7 - x^2$, and its width is $2x$.

Thus, the area $A$ of the rectangle is: $A = \text{width} \times \text{height} = 2x \cdot (7 - x^2)$ $A = 14x - 2x^3$

Step 2: Maximize the area

To find the $x$-value that maximizes the area, differentiate $A$ with respect to $x$ and set the derivative equal to 0: $\frac{dA}{dx} = 14 - 6x^2$ Set $\frac{dA}{dx} = 0$: $14 - 6x^2 = 0$ $6x^2 = 14 \quad \Rightarrow \quad x^2 = \frac{7}{3}$ $x = \pm \sqrt{\frac{7}{3}}$ We use the positive root $x = \sqrt{\frac{7}{3}}$ since $x$ represents a length.

Step 3: Verify maximum with the second derivative

The second derivative is: $\frac{d^2A}{dx^2} = -12x$ At $x = \sqrt{\frac{7}{3}}$, $\frac{d^2A}{dx^2} = -12\sqrt{\frac{7}{3}} < 0$, confirming a maximum.

Step 4: Dimensions of the rectangle

• Width: $2x = 2\sqrt{\frac{7}{3}} = \frac{2\sqrt{21}}{3}$
• Height: $y = 7 - x^2 = 7 - \frac{7}{3} = \frac{14}{3}$

Thus, the dimensions of the rectangle are:

• Width: $\frac{2\sqrt{21}}{3}$
• Height: $\frac{14}{3}$

---

Final Answer:

The dimensions of the rectangle of largest area are: $\text{Width: } \frac{2\sqrt{21}}{3}, \quad \text{Height: } \frac{14}{3}$

Would you like me to show a graphical representation or any additional details?

Follow-Up Questions:

1. How is symmetry of the parabola used in determining the rectangle's dimensions?
2. Why is the second derivative test important in optimization problems?
3. Can the same process be applied to any parabola $y = ax^2 + bx + c$?
4. How would the result change if the parabola was not symmetric about the $y$-axis?
5. What happens to the rectangle's dimensions if the vertex of the parabola shifts?

Tip:

In optimization problems, always check the second derivative to confirm whether you have a maximum or minimum value!