Difference between revisions of "2013 AMC 8 Problems/Problem 22"

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==Problem==
 
==Problem==
 
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Toothpicks are used to make a grid that is <math>60</math> toothpicks long and <math>32</math> toothpicks wide. How many toothpicks are used altogether?
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
 
  
 
<asy>
 
<asy>
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draw(corner,(5,0)--(35,0));
 
draw(corner,(5,0)--(35,0));
 
draw(corner,(0,-5)--(0,-35));
 
draw(corner,(0,-5)--(0,-35));
for (int i=0; i<3; ++i)
+
for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}}
{
 
for (int j=0; j>-2; --j)
 
{
 
if ((i-j)<3)
 
{
 
add(corner,(50i,50j));
 
}
 
}
 
}
 
 
draw((5,-100)--(45,-100));
 
draw((5,-100)--(45,-100));
 
draw((155,0)--(185,0),dotted+linewidth(2));
 
draw((155,0)--(185,0),dotted+linewidth(2));
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==Solution==
 
==Solution==
 
+
There are <math>61</math> vertical columns with a length of <math>32</math> toothpicks, and there are <math>33</math> horizontal rows with a length of <math>60</math> toothpicks. An effective way to verify this is to try a small case, i.e. a <math>2 \times 3</math> grid of toothpicks. Thus, our answer is <math>61\cdot 32 + 33 \cdot 60 = \boxed{\textbf{(E)}\ 3932}</math>.
There are <math>61</math> vertical columns with a length of <math>32</math> toothpicks, and there are <math>33</math> horizontal rows with a length of <math>60</math> toothpicks. An effective way to verify this is to try a small case, i.e. a <math>2 \times 3</math> grid of toothpicks. Thus, our answer is <math>61\cdot 32 + 33 \cdot 60 = \boxed{\textbf{(E)}\ 3952}</math>.
 
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2013|num-b=21|num-a=23}}
 
{{AMC8 box|year=2013|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 21:16, 11 February 2020

Problem

Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?

[asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}} draw((5,-100)--(45,-100)); draw((155,0)--(185,0),dotted+linewidth(2)); draw((105,-50)--(135,-50),dotted+linewidth(2)); draw((100,-55)--(100,-85),dotted+linewidth(2)); draw((55,-100)--(85,-100),dotted+linewidth(2)); draw((50,-105)--(50,-135),dotted+linewidth(2)); draw((0,-105)--(0,-135),dotted+linewidth(2));[/asy]

$\textbf{(A)}\ 1920 \qquad \textbf{(B)}\ 1952 \qquad \textbf{(C)}\ 1980 \qquad \textbf{(D)}\ 2013 \qquad \textbf{(E)}\ 3932$

Solution

There are $61$ vertical columns with a length of $32$ toothpicks, and there are $33$ horizontal rows with a length of $60$ toothpicks. An effective way to verify this is to try a small case, i.e. a $2 \times 3$ grid of toothpicks. Thus, our answer is $61\cdot 32 + 33 \cdot 60 = \boxed{\textbf{(E)}\ 3932}$.

See Also

2013 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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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