<p>This was a pretty hard geometric problem for me to conceptualize and visualize and collegeboard's explanation is trash like usual so i was hoping someone could help me:</p>
<p>page 548 #16 </p>
<p>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>2*(3)^1/2
draw down a picture and you will see that the diagonal of the cube is actually the diametet of the sphere.</p>
<p>can cc post a picture or i’ll write down for you and post it here since to type down square root is extremely hard?</p>
<p>2 square root of 2 is the answer</p>
<p>The cube’s volume is 8, therefore each of its sides is 2 (V = s^3).</p>
<p>It’s kinda hard to visualize, but the diameter of the sphere is the diagonal of the cube. Draw this out to see that.</p>
<p>Now, you can ignore the sphere completely and focus on finding the diagonal of the cube. The best way to do this is to visualize a right triangle with:
Side 1: height of the cube
Side 2: diagonal of the cube’s base
Hypotenuse: diagonal of the cube itself</p>
<p>Side 1 is easy, that’s just 2. Side 2 is more difficult. Side 2 is also the hypotenuse of a 45-45-90 right triangle with legs 2 and 2. Hence, Side 2 equals 2*sqrt(2) (the ratio of sides in a 45-45-90 triangle is 1:1:sqrt(2) ). Now we can use the Pythagorean theorem to find the third side:</p>
<p>(side 1)^2 + (side 2)^2 = (hypotenuse)^2
(2)^2 + (2sqrt(2))^2 = (hypotenuse)^2
4 + 4*2 = (hypotenuse)^2
12 = (hyptenuse)^2
hypotenuse = sqrt(12)
hypotenuse = 2sqrt(3)</p>
<p>This problem was pretty hard for me to visualize too the first time I came across it. So, I made a video to help everyone visualize this problem a little better. Check it out, here is the link: </p>
<p>[cube</a> inscribed in a sphere EXPLAINED with picture!!! - YouTube](<a href=“cube inscribed in a sphere EXPLAINED with picture!!! - YouTube”>cube inscribed in a sphere EXPLAINED with picture!!! - YouTube)</p>
<p>Quick solution: Use the Generalized Pythagorean Theorem: d^2 = a^2+b^2+c^2 = 2^2+2^2+2^2 = 3(2^2). So d = 2sqrt(3).</p>
<p>The Generalized Pythagorean Theorem can always be used to find the length of the LONG diagonal of a box. Another standard application is finding the diameter of a sphere with a cube inscribed in it.</p>
<p>Exercise: What if the sphere is inscribed in the cube instead?</p>