Question

he volume of a cone of radius r and height h is​ one-third the volume of a cylinder with the same radius and height. Does the surface area of a cone of radius r and height h equal​ one-third the surface area of a cylinder with the same radius and​ height? If​ not, find the correct relationship. Exclude the bases of the cone and cylinder.

Answer 1

Answer:

The surface area of a cone of radius r and height h not equal​ to one-third the surface area of a cylinder with the same radius and​ height.

Relationship is $S_c=(\frac{\sqrt{(r+h)}}{2h})S_C$

Step-by-step explanation:

Given : The volume of a cone of radius r and height h is​ one-third the volume of a cylinder with the same radius and height.

To find : Does the surface area of a cone of radius r and height h equal​ one-third the surface area of a cylinder with the same radius and​ height?

If​ not, find the correct relationship. Exclude the bases of the cone and cylinder.

Solution :

Radius of cone and cylinder is 'r'.

Height of cone and cylinder is 'h'.

The volume of cone is $V_c=\frac{1}{3}\pi r^2 h$

The volume of cylinder is $V_C=\pi r^2 h$

$\frac{V_c}{V_C}=\frac{\frac{1}{3}\pi r^2 h}{\pi r^2 h}$

$V_c=\frac{1}{3}V_C$

i.e. volume of cone is one-third of the volume of cylinder.

Now,

Surface area of the cone is $S_c=\pi r\sqrt{(r+h)}$

Surface area of the cylinder is $S_C=2\pi rh$

Dividing both the equations,

$\frac{S_c}{S_C}=\frac{\pi r\sqrt{(r+h)}}{2\pi rh}$

$\frac{S_c}{S_C}=\frac{\sqrt{(r+h)}}{2h}$

$S_c=(\frac{\sqrt{(r+h)}}{2h})S_C$

Which clearly means $S_c\neq \frac{1}{3}S_C$

i.e. The surface area of a cone of radius r and height h not equal​ to one-third the surface area of a cylinder with the same radius and​ height.

The relationship between them is

$S_c=(\frac{\sqrt{(r+h)}}{2h})S_C$