2018 AMC 10B Problems/Problem 10
Problem
In the rectangular parallelepiped shown, = , = , and = . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?
Solution 1
Consider the cross-sectional plane and label its area . Note that the volume of the triangular prism that encloses the pyramid is , and we want the rectangular pyramid that shares the base and height with the triangular prism. The volume of the pyramid is , so the answer is . (AOPS12142015)
Solution 2
We can start by finding the total volume of the parallelepiped. It is , because a rectangular parallelepiped is a rectangular prism.
Next, we can consider the wedge-shaped section made when the plane cuts the figure. We can find the volume of the triangular pyramid with base and apex . The area of is . Since is given to be , we have that is . Using the formula for the volume of a triangular pyramid, we have . Also, since the triangular pyramid with base and apex has the exact same dimensions, it has volume as well.
The original wedge we considered in the last step has volume , because it is half of the volume of the parallelepiped. We can subtract out the parts we found to have . Thus, the volume of the figure we are trying to find is . This means that the correct answer choice is .
Written by: Archimedes15
NOTE: For those who think that it isn't a rectangular prism, please read the problem. It says "rectangular parallelepiped." If a parallelepiped is such that all of the faces are rectangles, it is a rectangular prism.
Solution 3
If you look carefully, you will see that on the either side of the pyramid in question, there are two congruent tetrahedra. The volume of one is , with its base being half of one of the rectangular prism's faces and its height being half of one of the edges, so its volume is . We can obtain the answer by subtracting twice this value from the diagonal half prism, or
Solution 4
You can calculate the volume of the rectangular pyramid by using the formula, . is the area of the base, , and is equal to . The height, , is equal to the height of triangle drawn from to .
Area of
Area of (since Area ).
Area of
Volume of pyramid
Answer is
~OlutosinNGA
Solution 5
We can start by identifying the information we need. We need to find the area of rectangle and the height of rectangular prism .
In order to find the area of we can use the Pythagorean Theorem. We find that , so the area of rectangle . We shall refer to this as .
In order to find the height of rectangular prism , we can examine triangle . We can use the Geometric Mean Theorem to find that when an altitude is dropped from point is split into segments of length and . Taking the geometric mean of these numbers, we find that the altitude has length . This is also the height of the rectangular prism, which we shall refer to as .
Plugging and into the formula we find that the volume is . The answer is .
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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