Surface area of sphere = 4πr^2 =100 πcm^2
curved surface area of cone = πrl 4πl
ATQ, surface area of sphere=5× curved surface area of cone100π=4πl× 5100=20l5= slant height (l)
l^2=r^2+ h^25^2=4^2+h^225=16+ h^225—16=h^29=h^23=height
volume of cone= 1/3πr^2h = 352/7 50.28

The problem states that the surface area of a sphere with a radius of 5 cm is 5 times the curved surface area of a cone with the same radius. We need to find the height of the cone.

To solve this problem, let's break it down into smaller steps:

1. Find the surface area of the sphere:
The surface area of a sphere is given by the formula: 4πr^2, where r is the radius.
In this case, the radius is 5 cm, so the surface area of the sphere is:
Surface area of sphere = 4π(5^2) = 4π(25) = 100π cm^2.

2. Find the curved surface area of the cone:
The curved surface area of a cone is given by the formula: πrl, where r is the radius and l is the slant height.
In this case, the radius is also 5 cm. So, we need to find the slant height of the cone.

3. Relate the surface areas of the sphere and the cone:
The problem states that the surface area of the sphere is 5 times the curved surface area of the cone. Therefore, we can write the equation:
100π = 5(πrl).

4. Simplify the equation:
Divide both sides of the equation by π to eliminate it:
100 = 5rl.

5. Solve for the slant height:
We know that the radius of the cone is 5 cm. Let's substitute this value into the equation:
100 = 5(5)l.
100 = 25l.
Divide both sides of the equation by 25:
4 = l.

6. Find the height of the cone:
To find the height of the cone, we can use the Pythagorean theorem. The height, radius, and slant height form a right triangle.
Using the Pythagorean theorem, we have:
h^2 + 5^2 = 4^2.
h^2 + 25 = 16.
h^2 = 16 - 25.
h^2 = -9.

Since we cannot have a negative value for the height, it means that the problem does not have a valid solution. It is not possible to find the height of the cone in this case.

In conclusion, there is no solution to the problem. The height of the cone cannot be determined based on the given information.