Question

If the radii of a sphere, a cylinder and a cone are equal and their whole surface areas are equal, then the ratio of their heights is :

a. 2:1:2√2
b. √2:1:2
c. 2:1:3√2
d. 6√2:3√3:4

Answer 1

The ratio of the heights of the sphere, cylinder, and cone with equal radii and surface areas is 2:1:2√2.

Step-by-step explanation:

To find the ratio of heights of the spheres, cylinders, and cones, we can use the formulas for their surface areas and volumes. Let's assume the radius of each shape is 'r'.

The surface area of a sphere is given by 4πr², while the volume is (4/3)πr³.

For a cylinder, the surface area is 2πrh + 2πr², and the volume is πr²h.

And for a cone, the surface area is πr(r + √(r² + h²)), and the volume is (1/3)πr²h.

Given that the surface areas of all three shapes are equal, we can equate these formulas and solve for the ratio of heights.

After simplifying and solving the equation, we find that the ratio of heights is 2:1:2√2 (option a).