How many times will the volume of the cone increase if the radius of its base is increased

| August 1, 2021 | education

How many times will the volume of the cone increase if the radius of its base is increased by 4 times and the height is reduced by 2 times?

The formula for the volume of a cone is:

V = 1/3 * π * r ^ 2 * h.

Substitute into this formula the new values ​​of the radius increased by 4 times (r * 4) and the height decreased by 2 times (h / 2):

V1 = 1/3 * π * (r * 4) ^ 2 * (h / 2).

Let’s expand the brackets:

V1 = 1/3 * π * r ^ 2 * 4 ^ 2 * h * 1/2;

V1 = 1/3 * π * r ^ 2 * 16 * h * 1/2.

Or

V1 = 1/3 * π * r ^ 2 * h * 16 * 1/2;

V1 = 1/3 * π * r ^ 2 * h * 8.

Let’s find the ratio of the new volume value to the original one:

V1 / V = ​​(1/3 * π * r ^ 2 * h * 8) / (1/3 * π * r ^ 2 * h).

Reduce the fraction by (1/3 * π * r ^ 2 * h) and get:

V1 / V = ​​8.

This means that the new volume V1 is 8 times greater than the original V.

Therefore, if the radius of the base of the cone is increased by 4 times and the height is reduced by 2 times, then its volume will increase by 8 times.



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