Report on 11 November prize question

You have a perfectly spherical apple, and cut out the core, leaving a hole in the apple which is a perfect cylinder of height 6cm. What is the volume of apple left?

The year 12 further maths class submitted an answer as a group, and won the prize: 200 Vivos and a Freddo to each of them.

apple

The answer is 36π cm3.

If there is a definite answer to the question, then it must be the same whatever the size of cylinder, and therefore also the same if the cylinder is a pinhole of infinitesimal diameter and the apple has radius 3cm. In that case the apple’s volume is (4π/3).33, i.e. 36π, and no volume need be subtracted for the cylinder.

Why is the answer the same whatever the size of cylinder? The volume of apple left is the total of a lot of thin rings. Call the radius of the apple R, and the radius of the cylinder r. At height x above (or below) the centre of the apple, the area of the thin ring is, by Pythagoras, π(R2−r2−x2]). But if the height of the cylinder is 6, then, by Pythagoras again, R2−r2=9. So the area of the thin ring at height x is π(9−x2), and doesn’t depend on R and r!

And the number of thin rings must be same for big apples or small, since the height of the cylinder is 6cm.

Therefore the answer is the same whatever the size of cylinder. If you know about integration, you can get the answer as the integral from –3 to 3 of π(9–x2)dx.