Difference between revisions of "1996 AHSME Problems/Problem 17"
m (wikify) |
m (→Solution) |
||
(One intermediate revision by one other user not shown) | |||
Line 42: | Line 42: | ||
Then <math>x-2 = 6\sqrt{3} - 2</math>, and <math>y = \sqrt{3} (6 \sqrt{3} - 2) = 18 - 2\sqrt{3}</math>. | Then <math>x-2 = 6\sqrt{3} - 2</math>, and <math>y = \sqrt{3} (6 \sqrt{3} - 2) = 18 - 2\sqrt{3}</math>. | ||
− | The area of the | + | The area of the rectangle is thus <math>xy = 6\sqrt{3}(18 - 2\sqrt{3}) = 108\sqrt{3} - 36</math>. |
Using the approximation <math>\sqrt{3} \approx 1.7</math>, we get an area of just under <math>147.6</math>, which is closest to answer <math>\boxed{\text{E}}</math>. (The actual area is actually greater, since <math>\sqrt{3} > 1.7</math>). | Using the approximation <math>\sqrt{3} \approx 1.7</math>, we get an area of just under <math>147.6</math>, which is closest to answer <math>\boxed{\text{E}}</math>. (The actual area is actually greater, since <math>\sqrt{3} > 1.7</math>). | ||
Line 50: | Line 50: | ||
[[Category:Introductory Geometry Problems]] | [[Category:Introductory Geometry Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 22:45, 20 December 2016
Problem
In rectangle , angle is trisected by and , where is on , is on , and . Which of the following is closest to the area of the rectangle ?
Solution
Since , each of the three smaller angles is , and and are both triangles.
Defining the variables as illustrated above, we have from
Then , and .
The area of the rectangle is thus .
Using the approximation , we get an area of just under , which is closest to answer . (The actual area is actually greater, since ).
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.