2023 AMC 12B Problems/Problem 11
Revision as of 17:29, 15 November 2023 by Professorchenedu (talk | contribs)
Solution
Denote by 
the length of the shorten base.
Thus, the height of the trapezoid is
Thus, the area of the trapezoid is
$$ (Error compiling LaTeX. Unknown error_msg)
\begin{align*}
\frac{1}{2} \left( x + 2 x \right) \sqrt{1^2 - \left( \frac{x}{2} \right)^2}
& = \frac{3}{4} \sqrt{x^2 \left( 4 - x^2 \right)} \\
& \leq \frac{3}{4} \frac{x^2 + \left( 4 - x^2 \right)}{2} \\
& = \boxed{\textbf{(D) 
}} ,
\end{align*}
$$ (Error compiling LaTeX. Unknown error_msg)
where the inequality follows from the AM-GM inequality and it is binding if and only if 
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
