2020 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 20 Hard

Let B be a right rectangular prism (box) with edges lengths 1,3 , and 4 , together with its interior. For real r \geq 0, let S(r) be the set of points in 3 dimensional space that lie within a distance r of some point B. The volume of S(r) can be expressed as a r^{3}+b r^{2}+c r+d, where a, b, c, and d are positive real numbers. What is \frac{b c}{a d} ?(2020 AMC 10B, Question #20)

  • A.

    6

  • B.

    19

  • C.

    24

  • D.

    26

  • E.

    38

Answer:B

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 \left(\pi * r^{2} * h\right) / 4. The sum of all such cylinders must equal \left(\pi * r^{2}\right) / 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)}=(\mathbf{B}) 19

Related Problems
Problem 18 2020 AMC 10 B
Problem 19 2020 AMC 10 B
Problem 21 2020 AMC 10 B
Problem 22 2020 AMC 10 B
Think Academy Powered by ChatGPT