Difference between revisions of "2015 AMC 10A Problems/Problem 11"
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| − | + | ==Problem 11== | |
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| + | The ratio of the length to the width of a rectangle is <math>4</math> : <math>3</math>. If the rectangle has diagonal of length <math>d</math>, then the area may be expressed as <math>kd^2</math> for some constant <math>k</math>. What is <math>k</math>? | ||
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| + | <math>\textbf{(A)}\ \frac{2}{7}\qquad\textbf{(B)}\ \frac{3}{7}\qquad\textbf{(C)}\ \frac{12}{25}\qquad\textbf{(D)}}\ \frac{16}{25}\qquad\textbf{(E)}\ \frac{3}{4}</math> | ||
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| + | ==Solution== | ||
| + | Let the rectangle have length <math>4x</math> and width <math>3x</math>. Then by 3-4-5 triangles (or the Pythagorean Theorem), we have <math>d = 5x</math>, and so <math>x = \dfrac{x}{5}</math>. Hence, the area of the rectangle is <math>3x \cdot 4x = 12x^2 = \dfrac{12d^2}{25}</math>, so <math>k = \dfrac{12}{25} \textbf{(C)}</math>. | ||
Revision as of 18:21, 4 February 2015
Problem 11
The ratio of the length to the width of a rectangle is 
: 
. If the rectangle has diagonal of length 
, then the area may be expressed as 
for some constant 
. What is ?
$\textbf{(A)}\ \frac{2}{7}\qquad\textbf{(B)}\ \frac{3}{7}\qquad\textbf{(C)}\ \frac{12}{25}\qquad\textbf{(D)}}\ \frac{16}{25}\qquad\textbf{(E)}\ \frac{3}{4}$ (Error compiling LaTeX. Unknown error_msg)
Solution
Let the rectangle have length 
and width 
. Then by 3-4-5 triangles (or the Pythagorean Theorem), we have 
, and so 
. Hence, the area of the rectangle is 
, so 
.
