Work — Question 6

Question 6 – Solution

1. Choose a height coordinate and determine the radius.

Measure $y$ upward from the bottom vertex, so $0\le y\le5$. Similar triangles give$$\frac{r(y)}{y}=\frac25\quad\Longrightarrow\quad r(y)=\frac{2y}{5}.$$

2. Find the volume and weight of a slice.

$$dV=\pi r(y)^2dy=\frac{4\pi}{25}y^2dy,\qquad dF=9800\,dV=\frac{39200\pi}{25}y^2dy.$$

3. Include the lifting distance.

The outlet is at $y=5$, so a slice at $y$ rises $5-y$ meters:$$W=\frac{39200\pi}{25}\int_0^5 y^2(5-y)\,dy.$$

4. Expand and integrate.

$$y^2(5-y)=5y^2-y^3,\qquad W=\frac{39200\pi}{25}\left[\frac53y^3-\frac{y^4}{4}\right]_0^5.$$

5. Apply the bounds and simplify.

$$\begin{align*} W&=\frac{39200\pi}{25}\left(\frac{625}{3}-\frac{625}{4}\right)\\&=\frac{39200\pi}{25}\cdot\frac{625}{12}=\boxed{\frac{245000\pi}{3}\ \mathrm{J}}. \end{align*}$$