Difference between revisions of "2006 AMC 10B Problems/Problem 10"

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== Problem ==
 
== Problem ==
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?  
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In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is <math>15</math>. What is the greatest possible perimeter of the triangle?  
  
<math> \mathrm{(A) \ } 43\qquad \mathrm{(B) \ } 44\qquad \mathrm{(C) \ } 45\qquad \mathrm{(D) \ } 46\qquad \mathrm{(E) \ } 47 </math>
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<math> \textbf{(A) } 43\qquad \textbf{(B) } 44\qquad \textbf{(C) } 45\qquad \textbf{(D) } 46\qquad \textbf{(E) } 47 </math>
  
 
== Solution ==
 
== Solution ==

Revision as of 13:56, 26 January 2022

Problem

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $15$. What is the greatest possible perimeter of the triangle?

$\textbf{(A) } 43\qquad \textbf{(B) } 44\qquad \textbf{(C) } 45\qquad \textbf{(D) } 46\qquad \textbf{(E) } 47$

Solution

Let $x$ be the length of the first side.

The lengths of the sides are: $x$, $3x$, and $15$.

By the Triangle Inequality,

$3x < x + 15$

$2x < 15$

$x < \frac{15}{2}$

The greatest integer satisfying this inequality is $7$.

So the greatest possible perimeter is $7 + 3\cdot7 + 15 = 43 \Rightarrow A$

See Also

2006 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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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