Final answer:
The base of the solid is a circle with radius 3. The cross-sections perpendicular to the y-axis are equilateral triangles. The volume, (√3/4)V, can be found by integrating the area of the equilateral triangles along the y-axis.
Step-by-step explanation:
To find the volume of the solid, we need to integrate the area of the cross-sections along the y-axis. The equation of the base of the solid is x²+y²=9, which represents a circle with radius 3. The cross-sections perpendicular to the y-axis are equilateral triangles.
We can find the side length of the equilateral triangle by using the formula:
side length = 2 * (radius of the circle) * (sin(π/3))
side length = 2 * 3 * (sin(π/3)) = 6 * (sqrt(3)/2) = 3*sqrt(3)
The area of an equilateral triangle is given by:
Area = (sqrt(3)/4) * (side length)²
Area = (sqrt(3)/4) * (3*sqrt(3))² = 9*(sqrt(3)/4) = (9*sqrt(3))/4
Finally, we integrate the area of the cross-sections along the range of y-values to find the volume:
Volume = ∫[(9*sqrt(3))/4] dy = (9*sqrt(3)/4) ∫dy = (9*sqrt(3)/4) * (y)
Since the volume is given by (√3/4)V, we can equate the equation above to (√3/4)*V and solve for V:
√3/4 * V = (9*sqrt(3)/4) * (y)
V = (9/√3) * (y)