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Difference between revisions of "1986 AJHSME Problems/Problem 18"

m (→‎Solution)
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<asy>
 
<asy>
 +
unitsize(12);
 
draw((0,0)--(16,12));
 
draw((0,0)--(16,12));
−
draw((5.33333,4)--(10.66666,8)--(6.66666,13.33333)--(1.33333,9.33333)--cycle);
+
draw((10.66666,8)--(6.66666,13.33333)--(1.33333,9.33333)--(5.33333,4));
 
label("WALL",(7,4),SE);
 
label("WALL",(7,4),SE);
 
</asy>
 
</asy>
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==Solution==
 
==Solution==
  
−
The shortest possible rectangle that has sides 36 and 60 would be if the side opposite the wall was 80.  
+
The shortest possible rectangle that has sides 36 and 60 would be if the side opposite the wall was 60.  
  
 
Each of the sides of length 36 contribute <math>\frac{36}{12}+1=4</math> fence posts and the side of length 60 contributes <math>\frac{60}{12}+1=6</math> fence posts, so there are <math>4+4+6=14</math> fence posts.   
 
Each of the sides of length 36 contribute <math>\frac{36}{12}+1=4</math> fence posts and the side of length 60 contributes <math>\frac{60}{12}+1=6</math> fence posts, so there are <math>4+4+6=14</math> fence posts.   
  
−
However, the two corners where a 36 foot fence meets an 80 foot fence are counted twice, so there are actually <math>14-2=12</math> fence posts.
+
However, the two corners where a 36 foot fence meets an 60 foot fence are counted twice, so there are actually <math>14-2=12</math> fence posts.
  
 
<math>\boxed{\text{B}}</math>
 
<math>\boxed{\text{B}}</math>
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==See Also==
 
==See Also==
  
−
[[1986 AJHSME Problems]]
+
{{AJHSME box|year=1986|num-b=17|num-a=19}}
 +
[[Category:Introductory Geometry Problems]]

Revision as of 07:38, 23 May 2009

Problem

A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?

[asy] unitsize(12); draw((0,0)--(16,12)); draw((10.66666,8)--(6.66666,13.33333)--(1.33333,9.33333)--(5.33333,4)); label("WALL",(7,4),SE); [/asy]

$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

Solution

The shortest possible rectangle that has sides 36 and 60 would be if the side opposite the wall was 60.

Each of the sides of length 36 contribute $\frac{36}{12}+1=4$ fence posts and the side of length 60 contributes $\frac{60}{12}+1=6$ fence posts, so there are $4+4+6=14$ fence posts.

However, the two corners where a 36 foot fence meets an 60 foot fence are counted twice, so there are actually $14-2=12$ fence posts.

$\boxed{\text{B}}$

See Also

1986 AJHSME (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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