To solve this, we’ll start by calculating the surface area of the sphere and then use the relationship to determine the cone’s height and volume.

1. Surface Area of the Sphere:

The formula for the surface area of a sphere is:

$A_{\text{sphere}} = 4 \pi r^2$

For a sphere of radius $r = 5 \, \text{cm}$:

$A_{\text{sphere}} = 4 \pi (5)^2 = 100 \pi \, \text{cm}^2$

It is given that this is five times the curved surface area of the cone.

2. Curved Surface Area of the Cone:

The formula for the curved surface area of a cone is:

$A_{\text{cone, curved}} = \pi r l$

where:

• $r$ is the radius of the cone base ($r = 4 \, \text{cm}$),
• $l$ is the slant height of the cone.

Let the curved surface area of the cone be $x$. From the problem:

$100 \pi = 5 \cdot x \quad \Rightarrow \quad x = \frac{100 \pi}{5} = 20 \pi$

Thus, the curved surface area of the cone is $20 \pi \, \text{cm}^2$.

Substitute $A_{\text{cone, curved}} = 20 \pi$ into the formula:

$20 \pi = \pi (4) l \quad \Rightarrow \quad l = \frac{20 \pi}{4 \pi} = 5 \, \text{cm}$

So, the slant height of the cone is $l = 5 \, \text{cm}$.

3. Height of the Cone:

Using the Pythagorean theorem, we find the height of the cone ($h$):

$l^2 = r^2 + h^2 \quad \Rightarrow \quad 5^2 = 4^2 + h^2$

$25 = 16 + h^2 \quad \Rightarrow \quad h^2 = 9 \quad \Rightarrow \quad h = \sqrt{9} = 3 \, \text{cm}$

So, the height of the cone is $h = 3 \, \text{cm}$.

4. Volume of the Cone:

The formula for the volume of a cone is:

[ V = \frac{1}{3