Hello, fabxx!

If a square prism is inscribed in a right circular cylinder of radius 3 and height 6,

the volume inside the cylinder but outside the prism is:

. . $\displaystyle (a)\;2.14\qquad(b)\;3.14 \qquad (c)\;61.6 \qquad (d)\;115.6 \qquad (e)\;169.6$

Visualize a right circular cylinder (a soup can).

A square-based block is fitted into the can snugly.

They both have the same height, 6.

The base looks like this: Code:

* * *
* *
o - - - - - - - o
*| / |*
| / |
* | / | *
* | 6 / x| *
* | / | *
| / |
*| / x |*
o - - - - - - - o
* *
* * *

The diameter of the circle is 6.

Let $\displaystyle x$ = side of the square.

From Pythagorus: .$\displaystyle x^2+x^2\:=\:6^2 \quad\Rightarrow\quad 2x^2 \:=\:36 $

. . $\displaystyle x^2 \:=\:18\quad\Rightarrow\quad x \:=\:\sqrt{18} \:=\:3\sqrt{2}$

The volume is the cylinder is: .$\displaystyle V_1 \;=\;\pi r^2h \;=\;\pi(3^2)(6) \;=\;54\pi$

The volume of the block is: .$\displaystyle V_2 \;=\;LWH \;=\;(3\sqrt{2})(3\sqrt{2})(6) \;=\;108$

Solution: .$\displaystyle 54\pi - 108 \;\approx\;61.6$ . . . answer (c)