<p>pg. 548 of the blue book, # 16.</p>
<p>
[quote]
A cube with volume 8 cubic centimeters is inscribed in a sphere so that each vertex of the cube touches the sphere. What is the length of the diameter, in centimeters, of the sphere?</p>
<p>A)2
B)2.45
C)2.5
D)3.46
E)4
[/quote]
</p>
<p>The answer is D, but I can't figure out why...</p>
<p>Try using Pythagorean theorem two times. My S asked me this question some time back. I think you have to find the diagonal on one cube face, then use that information to find the distance between two opposite vertices.</p>
<p>The key to this problem is envisioning what the diameter of the sphere must look like in relation to the cube. Since the cube is inscribed in the sphere, each of the cube’s corners (vertices) will touch the sphere. The center of the sphere would be the center of the cube, and thus to find the diameter of the sphere, you have to find the diagonal of the cube.</p>
<p>To do this:</p>
<p>Find the diagonal of the base of the cube. Since the volume of the cube is 8, each side has a measure of 2 (2<em>2</em>2 = 8). </p>
<p>Now that you know the measure of each side, you can find the diagonal of the base of the cube using the Pythagorean theorem: (2)^2 + (2)^2 = root 8.</p>
<p>Now that you have the diagonal of the base of the cube AND you know the height of the cube is 2, you can find the diagonal of the cube using the Pythagorean theorem. </p>
<p>Root8 squared + 2 squared = root (8+4) = root 12. You can use a calculator to see that root 12 is equal to 3.46</p>