Question

A solid consists of a cylinder and a hemisphere of equal radius fixed base to base. Find theratio of the radius to the height of the cylinder, so that the solid has its centre of gravity atthe common face.​

Answer 1

This is the answer

Hope it helps

Answer 2

Tip:

• Surface Area of Cylinder $=2\pi rh$.
• Surface Area of Hemisphere $=2\pi r^2$

Explanation:

• A solid consists of a cylinder and a hemisphere of equal radius $(r)$ fixed base to base.
• Height of cylinder =$h$.
• We have to find the ratio of radius to the height of the cylinder.            

Steps:

Step 1 of 3:

Let the radius be $=r$

Height of the cylinder =$h$

Surface Area, $S=2\pi r^2+2 \pi rh+\pi r^2$

⇒    $\frac{S-3\pi r^2}{2\pi r}=h$               $\ldots (1)$

Volume, $V=\frac{2}{3}\pi r^3 +\pi r^2h$

⇒  $V= \frac{2}{3} \pi r^3 +\pi r^2\pi \left(\frac{S-3 \pi r^2}{2 \pi r}\right )$

⇒  $V=\frac{2}{3}\pi r^3+\frac{r}{2}S-\frac{3}{2}\pi r^3$

∴   $\frac{dV}{dr}=0$

⇒  $\frac{dV}{dr}=2\pi r^2+\frac{S}{2}-\frac{9}{2}\pi r^2 =0$

⇒  $\frac{S}{2}-\frac{5\pi r^2}{2}=0$

⇒  $S=5\pi r^2$

Putting the value of S in equation (1)

∴  $\frac{5\pi r^2-3\pi r^2}{2\pi r}=h$

∴  $\frac{h}{r}=1$

Final answer:

The ratio of the radius to the height of the cylinder is 1:1.