Difference between revisions of "2020 AMC 10B Problems/Problem 20"

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==Problem==
 
==Problem==
  
Let <math>B</math> be a right rectangular prism (box) with edges lengths <math>1,</math> <math>3,</math> and <math>4</math>, together with its interior. For real <math>r\geq0</math>, let <math>S(r)</math> be the set of points in <math>3</math>-dimensional space that lie within a distance <math>r</math> of some point <math>B</math>. The volume of <math>S(r)</math> can be expressed as <math>ar^{3} + br^{2} + cr +d</math>, where <math>a,</math> <math>b,</math> <math>c,</math> and <math>d</math> are positive real numbers. What is <math>\frac{bc}{ad}?</math>
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Let <math>B</math> be a right rectangular prism (box) with edges lengths <math>1,</math> <math>3,</math> and <math>4</math>, together with its interior. For real <math>r\geq0</math>, let <math>S(r)</math> be the set of points in <math>3</math>-dimensional space that lie within a distance <math>r</math> of some point in <math>B</math>. The volume of <math>S(r)</math> can be expressed as <math>ar^{3} + br^{2} + cr +d</math>, where <math>a,</math> <math>b,</math> <math>c,</math> and <math>d</math> are positive real numbers. What is <math>\frac{bc}{ad}?</math>
  
 
<math>\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38</math>
 
<math>\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38</math>
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==Solution==
 
==Solution==
  
Split the volume into 4 regions:  
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Split <math>S(r)</math> into 4 regions:  
  
 
1. The rectangular prism itself
 
1. The rectangular prism itself
  
2. The extensions of the faces of B
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2. The extensions of the faces of <math>B</math>
  
3. The quarter cylinders at each edge of B
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3. The quarter cylinders at each edge of <math>B</math>
  
4. The one-eighth spheres at each corner of B.
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4. The one-eighth spheres at each corner of <math>B</math>
  
Region 1: The volume of B is 12, so <math>d=12</math>
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Region 1: The volume of <math>B</math> is 12, so <math>d=12</math>
  
Region 2: The volume is equal to the surface area of B times r. The surface area can easily be computed to be <math>2(4*3 + 3*1 + 4*1) = 38</math>, so <math>c=38</math>.
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Region 2: The volume is equal to the surface area of <math>B</math> times <math>r</math>. The surface area can be computed to be <math>2(4*3 + 3*1 + 4*1) = 38</math>, so <math>c=38</math>.
  
Region 3: The volume of each quarter cylinder is equal to <math>(\pi*r^2*h)/4</math>. The sum of all such cylinders must equal <math>(\pi*r^2)/4</math> times the sum of the edge lengths. This can easily be computed as <math>4(4+3+1) = 32</math>, so the sum of the volumes of the quarter cylinders is <math>8\pi*r^2</math>. <math>b=8\pi</math>
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Region 3: The volume of each quarter cylinder is equal to <math>(\pi*r^2*h)/4</math>. The sum of all such cylinders must equal <math>(\pi*r^2)/4</math> times the sum of the edge lengths. This can be computed as <math>4(4+3+1) = 32</math>, so the sum of the volumes of the quarter cylinders is <math>8\pi*r^2</math>, so <math>b=8\pi</math>
  
Region 4: There is an eighth of a sphere of radius r at each corner. Since there are 8 corners, these add up to one full sphere of radius r. The volume of this sphere is <math>\frac{4}{3}\pi*r^3</math>. <math>a=\frac{4}{3}</math>.
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Region 4: There is an eighth of a sphere of radius <math>r</math> at each corner. Since there are 8 corners, these add up to one full sphere of radius <math>r</math>. The volume of this sphere is <math>\frac{4}{3}\pi*r^3</math>, so <math>a=\frac{4\pi}{3}</math>.
  
 
Using these values, <math>\frac{(8\pi)(38)}{(4\pi/3)(12)} = \boxed{\textbf{(B) }19}</math>
 
Using these values, <math>\frac{(8\pi)(38)}{(4\pi/3)(12)} = \boxed{\textbf{(B) }19}</math>
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~IceMatrix
 
~IceMatrix
 +
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=NAZTdSecBvs ~ MathEx
  
 
==See Also==
 
==See Also==

Revision as of 23:21, 13 July 2020

Problem

Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. What is $\frac{bc}{ad}?$

$\textbf{(A) } 6 \qquad\textbf{(B) } 19 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 26 \qquad\textbf{(E) } 38$

Solution

Split $S(r)$ into 4 regions:

1. The rectangular prism itself

2. The extensions of the faces of $B$

3. The quarter cylinders at each edge of $B$

4. The one-eighth spheres at each corner of $B$

Region 1: The volume of $B$ is 12, so $d=12$

Region 2: The volume is equal to the surface area of $B$ times $r$. The surface area can be computed to be $2(4*3 + 3*1 + 4*1) = 38$, so $c=38$.

Region 3: The volume of each quarter cylinder is equal to $(\pi*r^2*h)/4$. The sum of all such cylinders must equal $(\pi*r^2)/4$ times the sum of the edge lengths. This can be computed as $4(4+3+1) = 32$, so the sum of the volumes of the quarter cylinders is $8\pi*r^2$, so $b=8\pi$

Region 4: There is an eighth of a sphere of radius $r$ at each corner. Since there are 8 corners, these add up to one full sphere of radius $r$. The volume of this sphere is $\frac{4}{3}\pi*r^3$, so $a=\frac{4\pi}{3}$.

Using these values, $\frac{(8\pi)(38)}{(4\pi/3)(12)} = \boxed{\textbf{(B) }19}$

~DrJoyo

Video Solution

https://youtu.be/3BvJeZU3T-M

~IceMatrix

Video Solution

https://www.youtube.com/watch?v=NAZTdSecBvs ~ MathEx

See Also

2020 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 19
Followed by
Problem 21
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All AMC 10 Problems and Solutions

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