<p>A sphere is inscribed in a cube with a diagonal of 3(radical3) feet. In feet, what is the diameter of the sphere?</p>
<p>Answer is 3...</p>
<p>Hint: what is the side length of the cube, and how does this compare to the diameter of the inscribed sphere?</p>
<p>Wouldn’t they be equal?
So (3radical3)^2 = 2x^2
x = 3.67…</p>
<p>Just curious, how did you obtain the above equation?</p>
<p>If x is the side length of the cube, then applying Pythagorean twice, the diagonal of the cube has length x sqrt(3).</p>
<p>This is straight out of Cracking the ACT, 2013 Edition:</p>
<p>“The formula for the length of the diagonal of a rectangular prism is a^2+b^2+c^2 = d^2 where a, b and c represent the edges of the rectangular prism and d represents its diagonal. In the case of a cube, a = b = c, so the equation can be rewritten as follows: 3a^2 = d^2. In this problem, d = 3sqrt3, so 3a^2 = (3sqrt3)^2. Therefore, 3a^2 = (9)(3) and a = 3. Therefore, the length of the edge of this cube is 3.”"</p>
<p>Hope that helps. I find it weird that this book doesn’t cover the more difficult problems that can show up on the ACT rather than the basic stuff that everyone gets right anyway. The “formula for the length of the diagonal of a rectangular prism” doesn’t show up a single time in the math portion of the book.</p>