To find the dimensions of the rectangle with the greatest possible area, we approach this problem as an optimization problem.

Problem Breakdown:

1. Given:

   • The rectangle's base lies on the x-axis.
   • The upper corners of the rectangle touch the parabola $y = 10 - x^2$.
   • The width of the rectangle is $2x$ (as the base spans from $-x$ to $x$).
   • The height of the rectangle is $y = 10 - x^2$.

2. Area of the rectangle:

   • The area $A$ of the rectangle is given by: $A = \text{width} \times \text{height}$ Substituting: $A = 2x \cdot (10 - x^2)$ Simplify: $A = 20x - 2x^3$

3. Optimization:

   • To maximize the area, take the derivative of $A$ with respect to $x$ and set it to 0: $\frac{dA}{dx} = 20 - 6x^2$ Solve: $20 - 6x^2 = 0$ $6x^2 = 20 \quad \Rightarrow \quad x^2 = \frac{10}{3}$ x = \sqrt{\frac{10}{3}} \quad \text{(only positive values of x are valid)}

4. Calculate dimensions:

   • Width: $\text{Width} = 2x = 2\sqrt{\frac{10}{3}}$
   • Height: $\text{Height} = 10 - x^2 = 10 - \frac{10}{3} = \frac{20}{3}$

Final Dimensions:

• Width: $2\sqrt{\frac{10}{3}} \approx 3.65$
• Height: $\frac{20}{3} \approx 6.67$

Let me know if you would like further details or help with the calculations!

Related Questions:

1. How do you find the maximum area using the second derivative test?
2. Can this method be applied to other parabolas or curves?
3. What happens if the base of the rectangle is not on the x-axis?
4. How does the area change as $x$ approaches zero or the parabola's extremes?
5. What are the implications if the curve is a different function (e.g., sinusoidal)?

Tip:

Always simplify expressions as much as possible before taking derivatives to avoid errors in complex terms!