The height of a monument is m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side meters. Find the volume of the monument.

Question

The height of a monument is  m. A horizontal cross-section at a distance x meters from the top is an equilateral triangle with side  meters. Find the volume of the monument.

Quizplus · Accepted Answer

The base of the equilateral triangle at height $x$ is $11eaa3e5\_5588\_85eb\_9f8f\_ab033aa038ab\_TB5971\_11$ and its height is $\left(11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11 - x\right)$. Thus, the volume of the frustum with height $x$ is
$$\frac{1}{3}\pi\cdot\left(11eaa3e5\_5588\_85eb\_9f8f\_ab033aa038ab\_TB5971\_11\right)^2\cdot\left(11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11 - x\right) .$$
Integrating this expression from $x=0$ to $x=11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11$, we find the volume of the entire monument to be:
\begin{align*}
\int_0^{11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11} \frac{1}{3}\pi\cdot\left(11eaa3e5\_5588\_85eb\_9f8f\_ab033aa038ab\_TB5971\_11\right)^2\cdot\left(11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11 - x\right) dx &= \frac{1}{3}\pi\cdot\left(11eaa3e5\_5588\_85eb\_9f8f\_ab033aa038ab\_TB5971\_11\right)^2\cdot\int_0^{11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11} \left(11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11 - x\right) dx\
&= \frac{1}{3}\pi\cdot\left(11eaa3e5\_5588\_85eb\_9f8f\_ab033aa038ab\_TB5971\_11\right)^2\cdot\left[\frac{1}{2}\left(11eaa3e5\_5588\_5eda\_9f8f\_cd9ca38c3654\_TB5971\_11\right)^2\right]\
&=\boxed{11eaa3e5\_5588\_acfe\_9f8f\_4f0a3d7e8138\_TB5971\_11}
\end{align*}

The Height of a Monument Is M