Chapter 5, Problem 1P

To determine

To find:

Where the cuts are made on the pizza to obtain 3 same pizza pieces

Answer to Problem 1P

The cuts are made on $1.85$ inch distance from the center to get same 3 same pizza pieces.

Explanation of Solution

Given information:

The size of the pizza is 14 inch.

The pizza is decided to cut in parallel way.

Calculation:

Assume the radius of the pizza as $R$ which is 7 inch.

The distance of any general strip from the center be $r$ ,

So, the length of the strip is $2\sqrt{R^2-r^2}$

The width of the strip is $dr$

Now, the area of the strip $=2\sqrt{R^2-r^2}dr$

Let the distance of the slicing from the center be $d$ ,

Integrating from $d$ to $R$ the area of the slice can be obtained. But it is the one third of the total area, so it is $\frac{\pi R^2}{3}$

Thus, the share of each student can now be obtained as,

  $\begin{array}{l}\frac{\pi R^2}{3}=2{\displaystyle {\int }_d^R\sqrt{R^2-r^2}}dr\\ =R^2{\cos }^{-1}\frac{d}{R}-d\sqrt{R^2-r^2}=1.85\end{array}$

Therefore, the cuts are made on $1.85$ inch distance from the center to get same 3 same pizza pieces.