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> |
Revision as of 14:10, 20 March 2020
Contents
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
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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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