A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.
A) $\frac { \pi } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) $\frac { 3 \pi } { 2 }$
D) $\frac { 3 \sqrt { 3 } } { 2 }$
E) $\frac { 2 \pi } { 3 }$
F) $\frac { 4 \sqrt { 3 } } { 3 }$
G) $\frac { 3 \pi } { 4 }$
H) $\frac { 2 \sqrt { 2 } } { 3 }$

Question

A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.
A) $\frac { \pi } { 2 }$ 
B) $\frac { \sqrt { 3 } } { 2 }$ 
C) $\frac { 3 \pi } { 2 }$ 
D) $\frac { 3 \sqrt { 3 } } { 2 }$ 
E) $\frac { 2 \pi } { 3 }$ 
F) $\frac { 4 \sqrt { 3 } } { 3 }$ 
G) $\frac { 3 \pi } { 4 }$ 
H) $\frac { 2 \sqrt { 2 } } { 3 }$

Quizplus · Accepted Answer

The volume of the solid is found by integrating the area of the equilateral triangle cross-sections. The side length of the triangle at a distance $x$ from the center is $2\sqrt{1-x^2}$. The area of the triangle is $\frac{\sqrt{3}}{4}(2\sqrt{1-x^2})^2$. Integrating this from $-1$ to $1$ gives the volume $\frac{4\sqrt{3}}{3}$.

A Solid Has a Circular Base of Radius 1 π2\frac { \pi } { 2 }2π​

A Solid Has a Circular Base of Radius 1 $\frac { \pi } { 2 }$