For what radius length can the value of the volume of a sphere equal the value of the surface area?

When the sphere has a radius of length 3, the value of the volume of a sphere becomes equal to the value of the surface area of the same sphere.

Step-by-step explanation:

Given R is the radius of a sphere.

Volume of a sphere is given by the formula

/frac{4}{3} * π * R³.

While, surface area of a sphere is given by the formula

4 * π * R².

We need to find the value of radius, R, for which the value of the volume of a sphere equals the value of the surface are of a sphere.

As per the question, we begin with the assumption that the value of the volume of a sphere equals the value of the surface area of the sphere.

This is shown in the form of an equation given below.

/frac{4}{3} * π * R³ = 4 * π * R²

On both sides of the equation, π, is a common term. Hence, this term can be removed from both the sides.

⇒ / frac{4}{3} * R³ = 4 * R²

Similarly, R², is common on both sides of the equation. Next, this term is removed from both the sides of the above equation.

⇒ / frac{4}{3} * R = 4

Next, 4 is the numerator on both sides of the equation.

4 is removed in the next step from both sides.

We get the equation as shown below.

⇒ / frac{1}{3} * R = 1

In the above equation, only term left is R, the radius of the sphere.

On further solving the equation, we get the value of R as 3, which is shown below.

⇒ / frac{R}{3} = 1

On transferring 3 to the other side, we get the value of R as 3.

⇒ R = 1 x 3

⇒ R = 3

Hence, for the radius of 3, both the volume of the sphere and the surface area of the sphere become equal.