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 ?
![[asy] size(250); defaultpen(fontsize(10pt)); pair A =origin; pair B = (4.75,0); pair E1=(0,3); pair F = (4.75,3); pair G = (5.95,4.2); pair C = (5.95,1.2); pair D = (1.2,1.2); pair H= (1.2,4.2); pair M = ((4.75+5.95)/2,3.6); draw(E1--M--H--E1--A--B--E1--F--B--M--C--G--H); draw(B--C); draw(F--G); draw(A--D--H--C--D,dashed); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,E); label("$D$",D,W); label("$E$",E1,W); label("$F$",F,SW); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,N); dot(A); dot(B); dot(E1); dot(F); dot(G); dot(C); dot(D); dot(H); dot(M); label("3",A/2+B/2,S); label("2",C/2+G/2,E); label("1",C/2+B/2,SE); [/asy]](http://latex.artofproblemsolving.com/8/8/5/885c4e50ebdac89dd6aba698666a62eaa1b5eba6.png)
Solution 1
Consider the cross-sectional plane, and label it as b. Note that 
and we want 
, so the answer is 
. (AOPS12142015)
IMPORTANT: This is assuming the parallelepiped is a rectangular prism, which isn't correct. All we know is that each side is a parallelogram, so this solution doesn't work.
Solution 2
IMPORTANT: This solution assumed that the parallelepiped is a rectangular prism, which isn't correct. All we know is that each side is a parallelogram, so this solution didn't work. Sorry Adarshk.
Solution 3
IMPORTANT: This solution assumed that the parallelepiped is a rectangular prism, which isn't correct. All we know is that each side is a parallelogram, so this solution didn't work. Sorry Archimedes15.
Solution 4 (Vectors)
IMPORTANT: This solution assumed that the parallelepiped is a rectangular prism, which isn't correct. All we know is that each side is a parallelogram, so this solution didn't work. Sorry 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 ![$\dfrac{1}{3} \cdot [GFBC] \cdot GH= \dfrac{1}{3} \cdot 2 \cdot 3= \boxed{2}.$](http://latex.artofproblemsolving.com/3/d/d/3ddfc74ff85bc4e355105ddc879db698e651b190.png)
(MathloverMC)
See Also
| 2018 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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