Hey,

This is on page 19 # 32 of the Barron's SAT Math II Book.

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

a) 2.14
b) 3.14
c) 61.6
d) 115.6
e)169.6

Can you please show the steps cuz i really dont get this question. Thanks in advance!!!!

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)