Q.53 - Calculus

Question

Find the work described in each of Exercises 49–60. 

The work required to pump all of the water out of the top of an upright conical tank that is 10 feet high and has a radius of 4 feet at the top. 

Step-by-Step Solution

Answer

The work required to pump all of the water out of the top of an upright conical tank that is 10 feet high and has a radius of 4 feet at the top is 8319.9997$\mathrm{\pi }$ foot pounds. 

Step 1. Given information.

Here we have given,

The work required to pump all of the water out of the top of an upright conical tank that is 10 feet high and has a radius of 4 feet at the top.

Step 2. Concept used.

Formula for work is,

$W=Fd$

Similar triangles:

 if two triangles are similar, then their corresponding sides are in equal proportion.

Step 3. Explanation.

We have give,

The work required to pump all of the water out of the top of an upright conical tank that is 10 feet high and has a radius of 4 feet at the top. 

From the given information,

Height of the conical tank is 10 ft and radius is 4 ft.

Suppose that height of the slice is y ft and radius of the height is x ft.

Volume of the slice = ${\mathrm{\pi r}}^2\mathrm{dy}$

The weight density of water = $\omega =62.4$ pounds per cubic ft.

The vertical distance = 10 - y.

Using similar triangle properties,

$\frac{x}{y}=\frac{4}{10}$

$\Rightarrow x=\frac{4}{10}y$

Using, $W=Fd=\omega Vd$

Substituting value for the $\omega , V \mathrm{and} d$,

$W=64.2\times \mathrm{\pi }\times {\left(\frac{4}{10}y\right)}^2\times \mathrm{dy}\times (10-\mathrm{y})$

Hence, 

$$\begin{gathered} W={\int }_0^{10}62.4\times \mathrm{\pi }\times {\left(\frac{4}{10}y\right)}^2\times \mathrm{dy}\times (10-\mathrm{y}) \\ =9.984\mathrm{\pi }{\int }_0^{10}{\mathrm{y}}^2(10-\mathrm{y})\mathrm{dy} \\ =9.984\mathrm{\pi } \times 833.33333 \\ =8319.9997\mathrm{\pi } \end{gathered}$$

So W = 8319.9997$\mathrm{\pi }$ foot pounds.

Step 4. Conclusion.

Hence, the work required to pump all of the water out of the top of an upright conical tank that is 10 feet high and has a radius of 4 feet at the top is 8319.9997$\mathrm{\pi }$ foot pounds.