<p>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) 169.6
B) 3.14
c) 115.6
d) 2.14
e) 61.6</p>
<p>No clue how to do this just know the answer is E.</p>
<p>The volume of the right cylinder is (3.14)r^2h = 169.76</p>
<p>Now, draw a circle. </p>
<p>Your circle represents the top of the cylinder. You are looking down on the top of the cylinder.</p>
<p>Inscribe the biggest square that you can inside that circle.</p>
<p>Draw a triangle that includes the center and two of the four corners of the circle.</p>
<p>This is a right, isoceles triangle. Each of the two equal sides is equal to 3, the radius.</p>
<p>By the Pythagorean Theorem, the hypotenuse of your triangle is 3 (2^(1/2)).</p>
<p>The length of that hypotenuse, 3 (2^(1/2)), is the greatest possible length (and width, we are talking about a square) in the proposed square prism. </p>
<p>Of course, the height of the proposed square prism is 6, the height of the cylinder.</p>
<p>The volume of the square prism will be, of course, l x w x h.</p>
<p>Length is 3 (2^(1/2))
Width is also 3(2^(1/2))
Height is 6.</p>
<p>Volume of the inscribed square prism is lwh, which is, skipping a few steps, (6)(9)(2)=108.</p>
<p>The target answer is then 169.76 - 108 = (approx) 61.6.</p>
<p>good explanation above...but simple way to look at this would be to do the following:</p>
<p>Get volume of cylinder first: 54 pi. good?</p>
<p>Now get volume of prism (its not a square prism (aka cube) by the way, your wording is wrong. It is a RECTANGULAR prism)</p>
<p>To get this, you must get base by using special triangles. Knowing that the diameter of the cylinder base is 6, we have to use that as the hypt. of the 45:45:90 triangle. Thus getting 6/(radical 2). good?</p>
<p>We know the hieght is 6 from the question, so in in order to get volume of prism, do 6<em>[6/(radical 2)]</em>[6/(radical 2)] = 108. good?</p>
<p>now simply minus minus that volume from the cylinder volume.
(54 pi - 108) = 61.64</p>
<p>PS: Check my site out. <a href="http://www.satprogs.com%5B/url%5D">www.satprogs.com</a> and take a look at the gform program (relates to this problem) or for that matter any other program that you like.</p>
<p>merudh,
your product might be good, but is it such a cool idea to post an almost identical solution just to bring attention to your site?
(shucks! is not my post causing the same effect?)</p>
<p>As far as square prizm, some sources do define it as cube. </p>
<p>Most others say it's right prism with a square base (or even rectangular prism with two square faces - <a href="http://www.gifted.uconn.edu/projectm3/pdf/Level%204/Shapes%20Glossary.pdf%5B/url%5D">http://www.gifted.uconn.edu/projectm3/pdf/Level%204/Shapes%20Glossary.pdf</a>; those gifted! they see things so different).
In this case cube is a particular type of square prism.</p>
<p>To make things slightly more intriguing, square prism itself is now a special case of cuboid, aka rectangular prism, aka rectangular parallelepiped.</p>
<p>Who says math is not a precise science?</p>
<p>It's just an isolated incident that, according to the "New Oxford American English Dictionary", N.Amer. trapesoid and trapesium reverse their meanings in Brit.</p>
<p>Case closed.</p>
<p>
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When God is ambiguously defined
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