Difference between revisions of "2021 AMC 10A Problems/Problem 24"
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Revision as of 22:22, 11 February 2021
Contents
Problem 24
The interior of a quadrilateral is bounded by the graphs of 
and 
, where 
a positive real number. What is the area of this region in terms of , valid for all 
?
Solution
The conditions and give 
and 
or 
and 
. The slopes here are perpendicular, so the quadrilateral is a rectangle.
Plug in 
and graph it. We quickly see that the area is 
, so the answer can't be 
or 
by testing the values they give (test it!). Now plug in 
. We see using a ruler that the sides of the rectangle are about 
and 
. So the area is about 
. Testing 
we get 
which is clearly less than 
, so it is out. Testing 
we get 
which is near our answer, so we leave it. Testing 
we get 
, way less than , so it is out. So, the only plausible answer is 
~firebolt360
Video Solution by OmegaLearn (System of Equations and Shoelace Formula)
~ pi_is_3.14
See also
| 2021 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 23 |
Followed by Problem 25 | |
| 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. 
