Question

The base of a solid is a right triangle whose base side has length a and whose perpendicular side has length (1/2)a. Find the volume of the solid if cross sections perpendicular to the base of the triangle are semicircles.

Answer 1

Let the right triangle have base a and height (1/2)a. The area of the triangle is:

(1/2) * a * (1/2)a = (1/4)a²

If we take a cross section perpendicular to the base of the triangle, we get a semicircle with radius equal to the height of the triangle (since the semicircle is perpendicular to the base). The radius of the semicircle is (1/2)a, so its area is:

(1/2) * pi * ((1/2)a)² = (1/8)pi * a²

Therefore, the volume of the solid is given by integrating the area of the semicircles over the length of the base of the triangle:

V = ∫(1/8)pi * a² dx, where x goes from 0 to a

V = (1/8)pi * a² * x | from 0 to a

V = (1/8)pi * a² * a - (1/8)pi * a² * 0

V = (1/8)pi * a³

Therefore, the volume of the solid is (1/8)pi * a³.