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> | + | 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 | + | 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 | + | 2. The extensions of the faces of <math>B</math> |
− | 3. The quarter cylinders at each edge of B | + | 3. The quarter cylinders at each edge of <math>B</math> |
− | 4. The one-eighth spheres at each corner of B | + | 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> | + | 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 | + | 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 | + | 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> | + | 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)} = 19</math> | + | Using these values, <math>\frac{(8\pi)(38)}{(4\pi/3)(12)} = \boxed{\textbf{(B) }19}</math> |
~DrJoyo | ~DrJoyo | ||
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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 be a right rectangular prism (box) with edges lengths and , together with its interior. For real , let be the set of points in -dimensional space that lie within a distance of some point in . The volume of can be expressed as , where and are positive real numbers. What is
Solution
Split into 4 regions:
1. The rectangular prism itself
2. The extensions of the faces of
3. The quarter cylinders at each edge of
4. The one-eighth spheres at each corner of
Region 1: The volume of is 12, so
Region 2: The volume is equal to the surface area of times . The surface area can be computed to be , so .
Region 3: The volume of each quarter cylinder is equal to . The sum of all such cylinders must equal times the sum of the edge lengths. This can be computed as , so the sum of the volumes of the quarter cylinders is , so
Region 4: There is an eighth of a sphere of radius at each corner. Since there are 8 corners, these add up to one full sphere of radius . The volume of this sphere is , so .
Using these values,
~DrJoyo
Video Solution
~IceMatrix
Video Solution
https://www.youtube.com/watch?v=NAZTdSecBvs ~ MathEx
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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All AMC 10 Problems and Solutions |
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