Question

A triangle with coordinates $(2,0),(2,6),$ and $(6,0)$ is rotated about the line $x=2$. Calculate the volume, in cubic units, of the resultant solid.

• A. $16$
• B. $8\pi$
• C. $16\pi$
• D. $32\pi$
• E. $48\pi$

Answer 1

The correct option is D $32\pi$

When the points $(2,0),(2,6),(6,0)$ are joined, we get a right angled triangle.

When this triangle is rotated about the line $x=2$, the resultant solid is a cone with radius $4(=6-2)$ units and height equal to $6(=6-0)$ units.

The volume of cone is given by $\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \pi \times 4\times 4\times 6=32\pi$ sq units.