Difference between revisions of "2011 AMC 10B Problems/Problem 14"
m |
(→Solution 2 (Quick)) |
||
(3 intermediate revisions by 2 users not shown) | |||
Line 14: | Line 14: | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
The perimeter of a rectangle is <math>2 (a + b) = 2 (31) = \boxed{\textbf{(C)} 62}</math> | The perimeter of a rectangle is <math>2 (a + b) = 2 (31) = \boxed{\textbf{(C)} 62}</math> | ||
+ | |||
+ | == Solution 2 (Quick) == | ||
+ | |||
+ | We see the answer choices or the perimeter are integers. Therefore, the sides of the rectangle are most likely integers that satisfy <math>a^2+b^2=25^2</math>. In other words, <math>(a,b,25)</math> is a set of Pythagorean triples. Guessing and checking, we have <math>(7,24,25)</math> as the triplet, as the area is <math>7 \cdot 24 = 168</math> as requested. Therefore, the perimeter is <math>2(7+24)=\boxed{\textbf{(C)} 62}</math>. | ||
== See Also== | == See Also== | ||
{{AMC10 box|year=2011|ab=B|num-b=13|num-a=15}} | {{AMC10 box|year=2011|ab=B|num-b=13|num-a=15}} | ||
+ | {{MAA Notice}} |
Latest revision as of 11:05, 13 July 2021
Problem
A rectangular parking lot has a diagonal of meters and an area of square meters. In meters, what is the perimeter of the parking lot?
Solution
Let the sides of the rectangular parking lot be and . Then and . Add the two equations together, then factor. The perimeter of a rectangle is
Solution 2 (Quick)
We see the answer choices or the perimeter are integers. Therefore, the sides of the rectangle are most likely integers that satisfy . In other words, is a set of Pythagorean triples. Guessing and checking, we have as the triplet, as the area is as requested. Therefore, the perimeter is .
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.