Difference between revisions of "2013 AMC 8 Problems/Problem 18"
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Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | ||
− | + | <asy>import three; | |
+ | currentprojection=orthographic(-8,15,15); | ||
+ | triple A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P; | ||
+ | A = (0,0,0); | ||
+ | B = (0,10,0); | ||
+ | C = (12,10,0); | ||
+ | D = (12,0,0); | ||
+ | E = (0,0,5); | ||
+ | F = (0,10,5); | ||
+ | G = (12,10,5); | ||
+ | H = (12,0,5); | ||
+ | I = (1,1,1); | ||
+ | J = (1,9,1); | ||
+ | K = (11,9,1); | ||
+ | L = (11,1,1); | ||
+ | M = (1,1,5); | ||
+ | N = (1,9,5); | ||
+ | O = (11,9,5); | ||
+ | P = (11,1,5); | ||
+ | //outside box far | ||
+ | draw(surface(A--B--C--D--cycle),white,nolight); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | draw(surface(E--A--D--H--cycle),white,nolight); | ||
+ | draw(E--A--D--H--cycle); | ||
+ | draw(surface(D--C--G--H--cycle),white,nolight); | ||
+ | draw(D--C--G--H--cycle); | ||
+ | //inside box far | ||
+ | draw(surface(I--J--K--L--cycle),white,nolight); | ||
+ | draw(I--J--K--L--cycle); | ||
+ | draw(surface(I--L--P--M--cycle),white,nolight); | ||
+ | draw(I--L--P--M--cycle); | ||
+ | draw(surface(L--K--O--P--cycle),white,nolight); | ||
+ | draw(L--K--O--P--cycle); | ||
+ | //inside box near | ||
+ | draw(surface(I--J--N--M--cycle),white,nolight); | ||
+ | draw(I--J--N--M--cycle); | ||
+ | draw(surface(J--K--O--N--cycle),white,nolight); | ||
+ | draw(J--K--O--N--cycle); | ||
+ | //outside box near | ||
+ | draw(surface(A--B--F--E--cycle),white,nolight); | ||
+ | draw(A--B--F--E--cycle); | ||
+ | draw(surface(B--C--G--F--cycle),white,nolight); | ||
+ | draw(B--C--G--F--cycle); | ||
+ | //top | ||
+ | draw(surface(E--H--P--M--cycle),white,nolight); | ||
+ | draw(surface(E--M--N--F--cycle),white,nolight); | ||
+ | draw(surface(F--N--O--G--cycle),white,nolight); | ||
+ | draw(surface(O--G--H--P--cycle),white,nolight); | ||
+ | draw(M--N--O--P--cycle); | ||
+ | draw(E--F--G--H--cycle); | ||
+ | |||
+ | label("10",(A--B),SE); | ||
+ | label("12",(C--B),SW); | ||
+ | label("5",(F--B),W);</asy> | ||
--[[User:Arpanliku|Arpanliku]] 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC | --[[User:Arpanliku|Arpanliku]] 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC |
Revision as of 23:13, 27 March 2014
Contents
Problem
Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
--Arpanliku 16:22, 27 November 2013 (EST) Courtesy of Lord.of.AMC
Solution 1
There are cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there are cubes. Hence, the answer is .
Solution 2
We can just calculate the volume of the prism that was cut out of the original box. Each interior side of the fort will be feet shorter than each side of the outside. Since the floor is foot, the height will be feet. So the volume of the interior box is .
The volume of the original box is . Therefore, the number of blocks contained in the fort is .
See Also
2013 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.