2017 AMC 10A Problems/Problem 11
Problem
The region consisting of all points in three-dimensional space within units of line segment has volume . What is the length ?
Solution 1
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 units of a segment. It can be seen that 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 (the volume of our cylinder + the volume of our two hemispheres will equal ):
, where is equal to the length of our line segment.
Solving, we find that .
Solution 2
Because this is just a cylinder and "half spheres", and the radius is , the volume of the half spheres is . Since we also know that the volume of this whole thing is , we do to get as the volume of the cylinder. Thus the height is over the base, or , so our answer is
~Minor edit by virjoy2001
Diagram for Solution
http://i.imgur.com/cwNt293.png
Video Solution
https://youtu.be/s4vnGlwwHHw?t=190
See Also
2017 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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