2021 AMC 10A Problems/Problem 24
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