2018 AMC 10B Problems/Problem 10
In the rectangular parallelpiped 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. Note that and we want , so the answer is . (AOPS12142015)
Solution 2
We start by finding side of base by using the Pythagorean theorem on . Doing this, we get
Taking the square root of both sides of the equation, we get . We can then find the area of rectangle , noting that
Taking the vertical cross-sectional plane of the rectangular prism, we see that the distance from point to base is the same as the distance from point to side . Calling the point where the altitude from vertex touches side as point , we can easily find this altitude using the area of right , as
Multiplying both sides of the equation by 2 and substituting in known values, we get
Deducing that the altitude from vertex to base is and calling the point of intersection between the altitude and the base as point , we get the area of the rectangular pyramid to be
Written by: Adharshk
Solution 3
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 EFB and apex M. The area of EFB is . Since BC is given to be , we have that FM is . Using the formula for the volume of a triangular pyramid, we have . Also, since the triangular pyramid with base HGC and apex M 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
Solution 4 (Vectors)
By the Pythagorean theorem, . Because , the area of the base is . Now, we need to find the height.
Define as the midpoint of and as the midpoint of . Consider a vector coordinate system with origin with and axes parallel to and respectively (positive direction is towards , positive direction is towards , positive direction is towards ). Then, The dot product of and is the length of the projection of onto multiplied by the length of , so dividing the dot product of and by the length of should give the length of the projection of onto . Doing this calculation, we get that the length of the projection is . Notice that this projection onto is the same as projecting onto the plane.
Denote as the foot of the projection of onto . Then is right, so is a right triangle. Applying the Pythagorean theorem on and calling (which is actually the height of the pyramid) , we get . Therefore, .
Now since we have the base and the height of the pyramid, we can find its volume. , so the answer is .
Written by: SS4
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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