2017 AMC 10A Problems/Problem 11
Problem
The region consisting of all point in three-dimensional space within 3 units of line segment has volume 216. What is the length ?
In order to solve this problem, we must first visualize what the region contained looks like. We know that, in a three dimensional plane, the region consisting of all points within units of a point would be a sphere with radius . However, we need to find the region containing all points within 3 units of a segment. Therefore, our region is a cylinder with two hemispheres on either end. We know the volume of our region, so we set up the following equation:
$\frac{4\pi}{3}3^3+9\pix=216$ (Error compiling LaTeX. Unknown error_msg)
Where is equal to the length of our line segment.
We isolate . This comes out to be