Question

The lateral surface area of a given right circular cylinder is twice its volume. If the height of the cylinder is twice its base diameter, determine the volume of the largest sphere you can put on the cylinder. Determine also the difference of the volume of the cylinder and sphere. 

Answer 1

For a right circular cylinder of radius $r$ and height $h,$ the lateral area is $A=2\pi rh$ and the volume is $V=\pi r^2 h$

The lateral surface area of a given right circular cylinder is twice its volume

$$2\pi rh=2\pi r^2h=>r=1 unit$$

The height of the cylinder is twice its base diameter

$$h=4r=4 units$$

The volume of the largest sphere we can put on the cylinder is

$$V_{sphere}=\dfrac{4}{3}\pi r^3 =\dfrac{4}{3}\pi (1)^3 =\dfrac{4\pi}{3}({units}^3)$$

The volume of the cylinder is

$$V_{cyl}=\pi(1)^2(4)=4\pi ({units}^3)$$

The difference of the volume of the cylinder and sphere is

$$\Delta V=V_{cyl}-V_{sphere}=4\pi -\dfrac{4\pi}{3}$$

$$=\dfrac{8\pi}{3}({units}^3)$$