For the final post of 2011 we wanted to leave you with a puzzle to solve as you take some time off to relax:

## The “apple core” puzzle

Consider an apple of radius r. From the apple we remove the ‘core’ of height 2h. What volume of apple remains?

Please add your answers (and include your workings) as comments below. We’ll publish the answer in January.

Have a nice break and we’ll be back with fresh content in 2012 – but in the meantime, please check out the articles on here and the expert’s own blogs.

## 5 Comments

D kua

21/12/2011

volume of sphere S= 4/3*pi*r^3

volume of cylinder of height 2h C= 2h*pi*r'^2 where r' is the radius of the cylinder

Volume of spherical caps E=2*((r-h)pi*2*r')

Volume of remaining segment V= S - C-E

Volume of remaining segment V= (4/3*pi*r^3)-(2h*pi*r'^2)-2*((r-h)pi*2*r')

James M

22/12/2011

There is a much nicer solution (Hint: how does r' relate to r and h?)

James

Munky

22/12/2011

Firstly, as we are talking about a sphere and the hight is 2h, then h must = r. Once we know this, we can discard the importance of r. So if we take that as a given (ie. that the solution is independent of r) then the solution reduced would be...

drum roll...

(4/3)*(pi)*h^3

[if I am wrong, then I apologise unreservedly]

Tom Harris

10/01/2012

Note that h is the height when the core is removed. This will always be less than r, the radius of the apple.

Tom Harris

10/01/2012

Folks, as I came up with this puzzle in the run-up to Christmas, it's probably only fair that I share with you an elegant solution. It is, of course, possible to calculate the volume using traditional multi variable calculus. This will cause most people's eyes to glaze over... Over at Red Gate we believe in 'ingeniously simple' so let's see if there is a better way.

Rather than trying to calculate the volume directly, let's solve an easier problem. The volume of the shape shown in the diagram below is simple to calculate, but importantly has the same dimensions as the original problem posed.

V = 2 Pi * h * (r^2 - r'^2) (volume of a cylinder)

r'^2 = r^2 - h^ 2 (pythagoras)

so V = 2 * Pi * h^3

This volume must be proportional to the volume of the apple that remains as the dimensions of the 2 shapes are the same.

So volume of apple remains = K * Pi * h^3. We determine K by using the limit when h = r. In this case the volume left is the whole sphere as nothing is removed. Thus K = 4/3

Volume of apple remaining = 4/3 * Pi * h^3

Easy when you know how!

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