Q. 30 - Calculus

Question

Use optimization techniques to answer the questions in Exercises 25–30.
Find the volume of the largest cylinder that fits inside a sphere of radius $10$.

Step-by-Step Solution

Answer

The volume of the largest cylinder is  $V=\frac{4000\mathrm{\pi }}{3\sqrt{3}}$.

Step 1. Given Information.

Radius of circle is $10$.

Step 2. Form an equation.

From the given information, 

$$\begin{gathered} r^2=R^2-{\left(\frac{h}{2}\right)}^2 V \\ ={\mathrm{\pi r}}^2\mathrm{h} \\ =\mathrm{\pi }({\mathrm{R}}^2-{\left(\frac{\mathrm{h}}{2}\right)}^2)\mathrm{h} \end{gathered}$$

Step 3. Differentiate with respect to h.

On differentiating,

$$\begin{gathered} \frac{dV}{dh}=\mathrm{\pi }({\mathrm{hR}}^2-\frac{{\mathrm{h}}^2}{4}) \\ \mathrm{\pi }({\mathrm{hR}}^2-\frac{{\mathrm{h}}^2}{4})=0 \end{gathered}$$

Step 4. Find the maximum volume.

The second derivative of the volume with respect to h is negative is $h>0$ such that the volume is maximal at $h=h_0$.

$$\begin{gathered} V=\frac{4{\mathrm{\pi R}}^3}{3\sqrt{3}} \\ =\frac{4\mathrm{\pi }{\left(10\right)}^3}{3\sqrt{3}} \\ V=\frac{4000\mathrm{\pi }}{3\sqrt{3}} \end{gathered}$$