2018 AMC 10B Problems/Problem 10
Contents
Problem
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
Solution 5 (slicker method)
Rotate the rectangular pyramid so that rectangle is the base of our rectangular pyramid. Now our height becomes We know that the volume of our rectangular pyramid is
(MathloverMC)
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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