Difference between revisions of "2016 AMC 10B Problems/Problem 11"

(Created page with "==Problem== Carl decided to in his rectangular garden. He bought <math>20</math> fence posts, placed one on each of the four corners, and spaced out the rest evenly along the...")
 
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<math>\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512</math>
 
<math>\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512</math>
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==Solution==
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If the dimensions are <math>a\times b</math>, then one side will have <math>a+1</math> posts (including corners) and the other <math>b+1</math>.
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The total number of posts is <math>2(a+b)=20</math>.
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<asy>size(7cm);fill((0,0)--(5,0)--(5,7)--(0,7)--cycle,lightgreen);
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for(int i=0;i<5;++i)dot((i,0),red);for(int i=0;i<7;++i)dot((5,i),blue);dot((5,7));
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draw(arc((0,0),.5,-90,-270)--arc((4,0),.5,90,-90)--cycle,gray+dotted);
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draw(arc((5,0),.5,-180,0)--arc((5,6),.5,0,180)--cycle,gray+dotted);
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draw((0,-1)--(5,-1),Arrows);draw((6,0)--(6,7),Arrows);
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label("$a$",(0,-1)--(5,-1),S);label("$b$",(6,0)--(6,7),E);
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label("$a$",(1,1));label("$b$",(4,5));
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</asy>
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Solve the system <math>\begin{cases}b+1=2(a+1)\\a+b=10\end{cases}</math> to get <math>a=3,\ b=7</math>. Then the area is <math>4a\cdot 4b=336</math> which is <math>\mathbf{(B)}</math>.
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==See Also==
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{{AMC10 box|year=2016|ab=B|num-b=10|num-a=12}}
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{{MAA Notice}}

Revision as of 11:36, 21 February 2016

Problem

Carl decided to in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the corners, has twice as many posts as the shorter side, including the corners. What is the area, in square yards, of Carl’s garden?

$\textbf{(A)}\ 256\qquad\textbf{(B)}\ 336\qquad\textbf{(C)}\ 384\qquad\textbf{(D)}\ 448\qquad\textbf{(E)}\ 512$

Solution

If the dimensions are $a\times b$, then one side will have $a+1$ posts (including corners) and the other $b+1$.

The total number of posts is $2(a+b)=20$.

[asy]size(7cm);fill((0,0)--(5,0)--(5,7)--(0,7)--cycle,lightgreen); for(int i=0;i<5;++i)dot((i,0),red);for(int i=0;i<7;++i)dot((5,i),blue);dot((5,7)); draw(arc((0,0),.5,-90,-270)--arc((4,0),.5,90,-90)--cycle,gray+dotted); draw(arc((5,0),.5,-180,0)--arc((5,6),.5,0,180)--cycle,gray+dotted); draw((0,-1)--(5,-1),Arrows);draw((6,0)--(6,7),Arrows); label("$a$",(0,-1)--(5,-1),S);label("$b$",(6,0)--(6,7),E); label("$a$",(1,1));label("$b$",(4,5)); [/asy]

Solve the system $\begin{cases}b+1=2(a+1)\\a+b=10\end{cases}$ to get $a=3,\ b=7$. Then the area is $4a\cdot 4b=336$ which is $\mathbf{(B)}$.

See Also

2016 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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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