Difference between revisions of "2015 AMC 8 Problems/Problem 8"
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| + | ==Problem== | ||
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What is the smallest whole number larger than the perimeter of any triangle with a side of length <math> 5</math> and a side of length <math>19</math>? | What is the smallest whole number larger than the perimeter of any triangle with a side of length <math> 5</math> and a side of length <math>19</math>? | ||
<math>\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57</math> | <math>\textbf{(A) }24\qquad\textbf{(B) }29\qquad\textbf{(C) }43\qquad\textbf{(D) }48\qquad \textbf{(E) }57</math> | ||
| − | + | ==Solution== | |
| − | + | We know from the [[Triangle Inequality]] that the last side, <math>s</math>, fulfills <math>s<5+19=24</math>. Adding <math>5+19</math> to both sides of the inequality, we get <math>s+5+19<48</math>, and because <math>s+5+19</math> is the perimeter of our triangle, <math>\boxed{\textbf{(D)}\ 48}</math> is our answer. | |
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| + | ==Solution 2== | ||
| + | <math>s</math>, fulfills <math>s<5+19=24</math>, meaning that s<24. The greatest value of s can be 23. 23+5+19 is 47. 48 is our nearest answer choice so its <math>\boxed{\textbf{(D)}\ 48}</math>.-TheNerdWhoIsNerdy. | ||
| + | |||
| + | ==Video Solution (HOW TO THINK CRITICALLY!!!)== | ||
| + | https://youtu.be/rxXHhn5PB9w | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/zUiKAoX2D_Q | ||
| + | |||
| + | ~savannahsolver | ||
==See Also== | ==See Also== | ||
Latest revision as of 19:50, 15 September 2024
Contents
Problem
What is the smallest whole number larger than the perimeter of any triangle with a side of length 
and a side of length 
?
Solution
We know from the Triangle Inequality that the last side, 
, fulfills 
. Adding 
to both sides of the inequality, we get 
, and because 
is the perimeter of our triangle, 
is our answer.
Solution 2
, fulfills , meaning that s<24. The greatest value of s can be 23. 23+5+19 is 47. 48 is our nearest answer choice so its .-TheNerdWhoIsNerdy.
Video Solution (HOW TO THINK CRITICALLY!!!)
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
| 2015 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
