The height of a monument is $20 \mathrm {~m}$. A horizontal cross-section at a distance $x$ meters from the top is an equilateral triangle with side $\frac { x } { 4 }$ meters. Find the volume of the monument. Select the correct answer.
A) $\frac { 125 \sqrt { 3 } } { 3 } \mathrm {~m} ^ { 3 }$
B) $\frac { 125 \sqrt { 2 } } { 3 } \mathrm {~m} ^ { 3 }$
C) $\frac { \sqrt { 3 } } { 3 } \mathrm {~m} ^ { 3 }$
D) $\frac { 125 \sqrt { 2 } } { 2 } \mathrm {~m} ^ { 3 }$
E) $125 \sqrt { 3 } \mathrm {~m} ^ { 3 }$

Question

Quizplus · Accepted Answer

The volume of the monument is given by the integral of the area of the cross-section from $x = 0$ to $x = 20$. The area of the cross-section is given by $A = \frac { \sqrt { 3 } } { 4 } x ^ { 2 }$. Therefore, the volume of the monument is$$V = \int _ { 0 } ^ { 20 } \frac { \sqrt { 3 } } { 4 } x ^ { 2 } d x = \frac { \sqrt { 3 } } { 4 } \left[ \frac { x ^ { 3 } } { 3 } \right] _ { 0 } ^ { 20 } = \frac { \sqrt { 3 } } { 4 } \left( \frac { 8000 } { 3 } \right) = \frac { 125 \sqrt { 3 } } { 3 } \mathrm {~m} ^ { 3 }$$

The Height of a Monument Is 20 m20 \mathrm {~m}20 m a Horizontal Cross-Section at a Distance

The Height of a Monument Is $20 \mathrm {~m}$ a Horizontal Cross-Section at a Distance