Question

Three students order a pizza that they want to split evenly. They decide to cut it with 2 parallel lines. If pizza has a diameter 18 inches, at what point should they cut it to make sure the three parts are the same size?

Answer 1

Draw a circle of radius 9 centered at (0,0)

Work with just the top half. If it can be divided into 3 equal parts by two vertical lines, so will the whole circle.

In fact, just work with the upper-right quarter-circle. If it can be divided by one vertical line so that the left section is half as big as the right section, then by symmetry another line on the other side of the y-axis will do the job.

So, we want a line at x=a where

2∫[0,a]√(81-x^2) dx = ∫[a,9]√(81-x^2) dx

Now, ∫√(81-x^2) dx = 1/2(x√(81-x^2) + 81 arcsin(x/9))

2(a√(81-a^2)+ 81arcsin(a/9))-(0+0) = (0+81*π/2)-(a√(81-a^2)+81 arcsin(a/9))

3(a√(81-a^2) + 81arcsin(a/9)) - 81π/2 = 0

Graphing that reveals that a = 2.3844

So, two lines 2.3844 inches from the center will do the trick.