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Difference between revisions of "2018 AMC 10B Problems/Problem 10"

m (→‎Solution 2)
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==Solution 2==
 
==Solution 2==
−
We can start by finding the total volume of the parallelepiped. It is <math>2 \cdot 3 \cdot 1 = 6</math>, because a rectangular parallelepiped is a rectangular prism.
+
We start by finding side <math>\overline{BE}</math> of base <math>BCHE</math> by using the Pythagorean theorem on <math>\triangle ABE</math>. Doing this, we get
  
−
Next, we can consider the wedge-shaped section made when the plane <math>BCHE</math> cuts the figure. We can find the volume of the triangular pyramid with base EFB and apex M. The area of EFB is <math>\frac{1}{2} \cdot 2 \cdot 3 = 3</math>. Since BC is given to be <math>1</math>, we have that FM is <math>\frac{1}{2}</math>. Using the formula for the volume of a triangular pyramid, we have <math>V = \frac{1}{3} \cdot \frac{1}{2} \cdot 3 = \frac{1}{2}</math>. Also, since the triangular pyramid with base HGC and apex M has the exact same dimensions, it has volume <math>\frac{1}{2}</math> as well.
+
<cmath>\overline{BE}^2 = \overline{AB}^2 + \overline{AE}^2 = 3^2 + 2^2 = 11.</cmath>
  
−
The original wedge we considered in the last step has volume <math>3</math>, because it is half of the volume of the parallelepiped. We can subtract out the parts we found to have <math>3 - \frac{1}{2} \cdot 2 = 2</math>. Thus, the volume of the figure we are trying to find is <math>2</math>. This means that the correct answer choice is <math>\boxed{E}</math>.
+
Taking the square root of both sides of the equation, we get <math>\overline{BE} = \sqrt {11}</math>. We can then find the area of rectangle <math>BCHE</math>, noting that
 +
 
 +
<cmath>[BCHE] = \overline{BE} \cdot \overline{BC} = \sqrt {11} \cdot 1 = \sqrt {11}.</cmath>
 +
 
 +
 
 +
Taking the vertical cross-sectional plane of the rectangular prism, we see that the distance from point <math>M</math> to base <math>BCHE</math> is the same as the distance from point <math>F</math> to side <math>\overline{BE}</math>. Calling the point where the altitude from vertex <math>F</math> touches side <math>\overline{BE}</math> as point <math>K</math>, we can easily find this altitude using the area of right <math>\triangle BFE</math>, as
 +
 
 +
<cmath>\frac{\overline{BF} \cdot \overline{FE}}{2} = \frac{\overline{BE} \cdot FK}{2}.</cmath>
 +
 
 +
Multiplying both sides of the equation by 2 and substituting in known values, we get
 +
 
 +
<math></math>2 \cdot 3 = FK \cdot \sqrt {11} \Rightarrow FK = \frac{6\sqrt {11}{11}.<math>
 +
 
 +
Deducing that the altitude from vertex </math>M<math> to base </math>BCHE<math> is </math>\frac{6\sqrt {11}{11}<math> and calling the point of intersection between the altitude and the base as point </math>N<math>, we get the area of the rectangular pyramid to be
 +
 
 +
<cmath>\frac{1}{3}([BCHE] \cdot MN) = \frac{1}{3}\left(\sqrt {11} \cdot \frac{6\sqrt {11}{11}\right) = \frac{66}{33} = \boxed{2}.</cmath>
 +
 
 +
Written by: Adharshk
 +
 
 +
 
 +
==Solution 3==
 +
We can start by finding the total volume of the parallelepiped. It is </math>2 \cdot 3 \cdot 1 = 6<math>, because a rectangular parallelepiped is a rectangular prism.
 +
 
 +
Next, we can consider the wedge-shaped section made when the plane </math>BCHE<math> cuts the figure. We can find the volume of the triangular pyramid with base EFB and apex M. The area of EFB is </math>\frac{1}{2} \cdot 2 \cdot 3 = 3<math>. Since BC is given to be </math>1<math>, we have that FM is </math>\frac{1}{2}<math>. Using the formula for the volume of a triangular pyramid, we have </math>V = \frac{1}{3} \cdot \frac{1}{2} \cdot 3 = \frac{1}{2}<math>. Also, since the triangular pyramid with base HGC and apex M has the exact same dimensions, it has volume </math>\frac{1}{2}<math> as well.
 +
 
 +
The original wedge we considered in the last step has volume </math>3<math>, because it is half of the volume of the parallelepiped. We can subtract out the parts we found to have </math>3 - \frac{1}{2} \cdot 2 = 2<math>. Thus, the volume of the figure we are trying to find is </math>2<math>. This means that the correct answer choice is </math>\boxed{E}$.
  
 
Written by: Archimedes15
 
Written by: Archimedes15

Revision as of 18:29, 16 February 2018

In the rectangular parallelpiped shown, $AB$ = $3$, $BC$ = $1$, and $CG$ = $2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$?


[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);[/asy]


$\textbf{(A) } 1 \qquad \textbf{(B) } \frac{4}{3} \qquad \textbf{(C) } \frac{3}{2} \qquad \textbf{(D) } \frac{5}{3} \qquad \textbf{(E) } 2$

Solution 1

Consider the cross-sectional plane. Note that $bh/2=3$ and we want $bh/3$, so the answer is $\boxed{2}$. (AOPS12142015)

Solution 2

We start by finding side $\overline{BE}$ of base $BCHE$ by using the Pythagorean theorem on $\triangle ABE$. Doing this, we get

\[\overline{BE}^2 = \overline{AB}^2 + \overline{AE}^2 = 3^2 + 2^2 = 11.\]

Taking the square root of both sides of the equation, we get $\overline{BE} = \sqrt {11}$. We can then find the area of rectangle $BCHE$, noting that

\[[BCHE] = \overline{BE} \cdot \overline{BC} = \sqrt {11} \cdot 1 = \sqrt {11}.\]


Taking the vertical cross-sectional plane of the rectangular prism, we see that the distance from point $M$ to base $BCHE$ is the same as the distance from point $F$ to side $\overline{BE}$. Calling the point where the altitude from vertex $F$ touches side $\overline{BE}$ as point $K$, we can easily find this altitude using the area of right $\triangle BFE$, as

\[\frac{\overline{BF} \cdot \overline{FE}}{2} = \frac{\overline{BE} \cdot FK}{2}.\]

Multiplying both sides of the equation by 2 and substituting in known values, we get

$$ (Error compiling LaTeX. Unknown error_msg)2 \cdot 3 = FK \cdot \sqrt {11} \Rightarrow FK = \frac{6\sqrt {11}{11}.$Deducing that the altitude from vertex$M$to base$BCHE$is$\frac{6\sqrt {11}{11}$and calling the point of intersection between the altitude and the base as point$N$, we get the area of the rectangular pyramid to be

<cmath>\frac{1}{3}([BCHE] \cdot MN) = \frac{1}{3}\left(\sqrt {11} \cdot \frac{6\sqrt {11}{11}\right) = \frac{66}{33} = \boxed{2}.</cmath>

Written by: Adharshk


==Solution 3== We can start by finding the total volume of the parallelepiped. It is$ (Error compiling LaTeX. Unknown error_msg)2 \cdot 3 \cdot 1 = 6$, because a rectangular parallelepiped is a rectangular prism.

Next, we can consider the wedge-shaped section made when the plane$ (Error compiling LaTeX. Unknown error_msg)BCHE$cuts the figure. We can find the volume of the triangular pyramid with base EFB and apex M. The area of EFB is$\frac{1}{2} \cdot 2 \cdot 3 = 3$. Since BC is given to be$1$, we have that FM is$\frac{1}{2}$. Using the formula for the volume of a triangular pyramid, we have$V = \frac{1}{3} \cdot \frac{1}{2} \cdot 3 = \frac{1}{2}$. Also, since the triangular pyramid with base HGC and apex M has the exact same dimensions, it has volume$\frac{1}{2}$as well.

The original wedge we considered in the last step has volume$ (Error compiling LaTeX. Unknown error_msg)3$, because it is half of the volume of the parallelepiped. We can subtract out the parts we found to have$3 - \frac{1}{2} \cdot 2 = 2$. Thus, the volume of the figure we are trying to find is$2$. This means that the correct answer choice is$\boxed{E}$.

Written by: Archimedes15

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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