Question

A manufacturer drills a hole through the center of a metal sphere of radius \(R\). The hole has a radius \(r\). Find the volume of the resulting ring.

Short answer

The volume of the resulting ring is given by \( V_ring = \frac{4}{3} \pi R^3 - \frac{4}{3} \pi r^3\).

Step-by-step solution

Compute the volume of the original sphere

Use the formula for the volume of a sphere which is given by \( V = \frac{4}{3} \pi R^3\) where R is the radius of the original sphere.

Compute the volume of the drilled sphere

The drilled sphere can also be regarded as a smaller sphere taken out of the bigger one. Thus, the volume of the drilled sphere can be calculated by using the same formula as in Step 1, substituting r for R. Hence the volume of the drilled sphere is \( V = \frac{4}{3} \pi r^3\)

Calculate the volume of the resulting ring

The volume of the resulting ring is attained by subtracting the volume of the drilled sphere from the original sphere. Mathematically, this can be represent as: \( V_ring = V_{sphere} - V_{drilled} = \frac{4}{3} \pi R^3 - \frac{4}{3} \pi r^3\).