This problem asks us to find the volume of a solid with a base bounded by the circle $x^2 + y^2 = 9$, where cross-sections perpendicular to the $x$-axis are equilateral triangles.

Step-by-step Solution:

1. Interpret the base: The circle $x^2 + y^2 = 9$ describes a circle centered at the origin with a radius of 3. Solving for $y$, we have: $y = \pm\sqrt{9 - x^2}.$ The base of the solid lies between $x = -3$ and $x = 3$.

2. Cross-section details: For a given $x$, the base of the equilateral triangle is the length of the vertical line segment determined by the circle. This length is: $2\sqrt{9 - x^2}.$ This represents the side length of the equilateral triangle.

3. Area of the equilateral triangle: The area of an equilateral triangle with side length $s$ is given by: $\text{Area} = \frac{\sqrt{3}}{4}s^2.$ Substituting $s = 2\sqrt{9 - x^2}$: $\text{Area} = \frac{\sqrt{3}}{4}\left(2\sqrt{9 - x^2}\right)^2 = \frac{\sqrt{3}}{4} \cdot 4(9 - x^2) = \sqrt{3}(9 - x^2).$

4. Volume of the solid: To find the total volume, integrate the area of the cross-sections along the $x$-axis from $-3$ to $3$: $V = \int_{-3}^{3} \sqrt{3}(9 - x^2) \, dx.$

5. Simplify the integral: Factor out $\sqrt{3}$: $V = \sqrt{3} \int_{-3}^{3} (9 - x^2) \, dx.$ Split the integral: $V = \sqrt{3} \left[\int_{-3}^{3} 9 \, dx - \int_{-3}^{3} x^2 \, dx\right].$

6. Evaluate each integral:

   • For $\int_{-3}^{3} 9 \, dx$: $\int_{-3}^{3} 9 \, dx = 9[x]_{-3}^{3} = 9(3 - (-3)) = 9(6) = 54.$

   • For $\int_{-3}^{3} x^2 \, dx$, note that $x^2$ is an even function, so: $\int_{-3}^{3} x^2 \, dx = 2\int_{0}^{3} x^2 \, dx.$ Compute: $\int_{0}^{3} x^2 \, dx = \left[\frac{x^3}{3}\right]_{0}^{3} = \frac{3^3}{3} - \frac{0^3}{3} = \frac{27}{3} = 9.$ Thus: $\int_{-3}^{3} x^2 \, dx = 2(9) = 18.$

7. Combine results: Substituting back: $V = \sqrt{3} \left(54 - 18\right) = \sqrt{3}(36).$

8. Final volume: $V = 36\sqrt{3}.$

Final Answer:

The volume of the solid is: $\boxed{36\sqrt{3}}$

Would you like further clarification or a graphical explanation? Let me know!

5 Related Questions:

1. What is the general formula for calculating volumes with cross-sections perpendicular to an axis?
2. How does the volume change if the cross-sections are squares instead of equilateral triangles?
3. What happens to the volume if the base circle's radius is doubled?
4. Can the same method be applied if the base region is bounded by an ellipse instead of a circle?
5. How do we calculate the volume if cross-sections are semicircles instead of triangles?

Tip:

Always check the symmetry of the problem—it can simplify calculations significantly, especially for definite integrals!