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> | ||
<math>\textbf{(A)}\ 204 \qquad \textbf{(B)}\ 280 \qquad \textbf{(C)}\ 320 \qquad \textbf{(D)}\ 340 \qquad \textbf{(E)}\ 600</math> | <math>\textbf{(A)}\ 204 \qquad \textbf{(B)}\ 280 \qquad \textbf{(C)}\ 320 \qquad \textbf{(D)}\ 340 \qquad \textbf{(E)}\ 600</math> | ||
− | ==Solution== | + | ==Solution 1== |
There are <math>10 \cdot 12 = 120</math> 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 <math>9 + 11 + 9 + 11 = 40</math> cubes. Hence, the answer is <math>120 + 4 \cdot 40 = \boxed{\textbf{(B)}\ 280}</math>. | There are <math>10 \cdot 12 = 120</math> 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 <math>9 + 11 + 9 + 11 = 40</math> cubes. Hence, the answer is <math>120 + 4 \cdot 40 = \boxed{\textbf{(B)}\ 280}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | We can just calculate the volume of the prism that was cut out of the original <math>12\times 10\times 5</math> box. Each interior side of the fort will be <math>2</math> feet shorter than each side of the outside. Since the floor is <math>1</math> foot, the height will be <math>4</math> feet. So the volume of the interior box is <math>10\times 8\times 4=320\text{ ft}^3</math>. | ||
+ | |||
+ | The volume of the original box is <math>12\times 10\times 5=600\text{ ft}^3</math>. Therefore, the number of blocks contained in the fort is <math>600-320=\boxed{\textbf{(B)}\ 280}</math>. | ||
+ | |||
+ | == Video Solution == | ||
+ | https://youtu.be/FDgcLW4frg8?t=5261 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==See Also== | ==See Also== | ||
+ | (Other problems) | ||
{{AMC8 box|year=2013|num-b=17|num-a=19}} | {{AMC8 box|year=2013|num-b=17|num-a=19}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 05:01, 28 January 2021
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?
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 .
Video Solution
https://youtu.be/FDgcLW4frg8?t=5261
~ pi_is_3.14
See Also
(Other problems)
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 |
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