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Difference between revisions of "2009 AMC 8 Problems/Problem 25"

(Created page with "==Problem== A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cube is <math> \tfrac12 </math> foot from the top face. ...")
 
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==Problem==
 
==Problem==
  
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A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cube is <math> \tfrac12 </math>
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A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is <math>\frac{1}{2}</math> foot from the top face. The second cut is <math>\frac{1}{3}</math> foot below the first cut, and the third cut is <math>\frac{1}{17}</math> foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
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foot from the top face. The second cut is <math> \tfrac13 </math> foot below the first cut, and the third cut is <math> \tfrac1{17} </math> foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
 
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<asy>
 
<asy>
 
import three;
 
import three;
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<math> \textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11 </math>
 
<math> \textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11 </math>
  
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==Solution==
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==Solution 1==
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The areas of the tops of <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> in the figure formed has sum <math> 1+1+1+1 = 4 </math> as do the bottoms. Thus, the total so far is <math>8</math>. Now, one of the sides has an area of one, since it combines all of the heights of <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, which is <math>1</math>. The other side is also the same. Thus the total area now is <math>10</math>. From the front, the surface area is half, because if you looked at it straight from the front it would look exactly like a side of <math>A</math>, with a surface area of half. From the back, it is the same thing. Thus, the total surface area is <math> 10+\frac{1}{2}+\frac{1}{2}= 11 </math>, or <math> \boxed{\textbf{(E)}\:11 } </math>.
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==Solution 2 (Faster method that relies on the MCQ choices)==
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The top parts and the bottom parts sum to 8. The sides add on another 2. Therefore, the only logical answer would be <math>\boxed{\textbf{(E)}\:11 }</math> since it is the only answer greater than 10.
  
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The tops of <math>A</math>, <math>B</math>, <math>C</math>,<math>D</math> , and <math>E</math> in the figure formed has sum <math> 1+1+1+1 = 4 </math>as do the bottoms. Thus, the total so far is <math>8</math>. Now, one of the sides has area one, since it combines all of the heights of <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>E</math>, which is <math>1</math>. The other side also satisfies this. Thus the total area now is <math>10</math>. From the front, the surface are is half, because if you looked at it straight from the front it would look exactly like <math>A</math>, with surface area half. From the back it is the same thing. Thus, the total is <math> 10+\frac{1}{2}+\frac{1}{2}= 11 </math>, or <math> \boxed{\bf{(E)}} </math>.
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==See Also==
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{{AMC8 box|year=2009|num-b=24|after=Last Problem}}
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{{MAA Notice}}

Latest revision as of 02:43, 31 October 2020

Problem

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled A, B, C, and D. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet? [asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(1,8/15,7/15); draw(unitcube, white, thick(), nolight); void f(real x) { draw((0,1,x)--(1,1,x)--(1,0,x)); } f(d); f(1/6); f(1/2); label("A", (1,0,3/4), W); label("B", (1,0,1/3), W); label("C", (1,0,1/6-d/4), W); label("D", (1,0,d/2), W); label("1/2", (1,1,3/4), E); label("1/3", (1,1,1/3), E); label("1/17", (0,1,1/6-d/4), E);[/asy]

[asy] import three; real d=11/102; defaultpen(fontsize(8)); defaultpen(linewidth(0.8)); currentprojection=orthographic(2,8/15,7/15); int t=0; void f(real x) { path3 r=(t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--cycle; path3 f=(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)--cycle; path3 u=(t,1,x)--(t+1,1,x)--(t+1,0,x)--(t,0,x)--cycle; draw(surface(r), white, nolight); draw(surface(f), white, nolight); draw(surface(u), white, nolight); draw((t,1,x)--(t+1,1,x)--(t+1,1,0)--(t,1,0)--(t,1,x)--(t,0,x)--(t+1,0,x)--(t+1,1,x)--(t+1,1,0)--(t+1,0,0)--(t+1,0,x)); t=t+1; } f(d); f(1/2); f(1/3); f(1/17); label("D", (1/2, 1, 0), SE); label("A", (1+1/2, 1, 0), SE); label("B", (2+1/2, 1, 0), SE); label("C", (3+1/2, 1, 0), SE);[/asy]

$\textbf{(A)}\:6\qquad\textbf{(B)}\:7\qquad\textbf{(C)}\:\frac{419}{51}\qquad\textbf{(D)}\:\frac{158}{17}\qquad\textbf{(E)}\:11$

Solution 1

The areas of the tops of $A$, $B$, $C$, and $D$ in the figure formed has sum $1+1+1+1 = 4$ as do the bottoms. Thus, the total so far is $8$. Now, one of the sides has an area of one, since it combines all of the heights of $A$, $B$, $C$, and $D$, which is $1$. The other side is also the same. Thus the total area now is $10$. From the front, the surface area is half, because if you looked at it straight from the front it would look exactly like a side of $A$, with a surface area of half. From the back, it is the same thing. Thus, the total surface area is $10+\frac{1}{2}+\frac{1}{2}= 11$, or $\boxed{\textbf{(E)}\:11 }$.

Solution 2 (Faster method that relies on the MCQ choices)

The top parts and the bottom parts sum to 8. The sides add on another 2. Therefore, the only logical answer would be $\boxed{\textbf{(E)}\:11 }$ since it is the only answer greater than 10.

See Also

2009 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
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 AJHSME/AMC 8 Problems and Solutions

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