Difference between revisions of "2023 AMC 12B Problems/Problem 11"

Line 10: Line 10:
  
 
Thus, the area of the trapezoid is
 
Thus, the area of the trapezoid is
<math></math>
+
<cmath>
 
\begin{align*}
 
\begin{align*}
 
\frac{1}{2} \left( x + 2 x \right) \sqrt{1^2 - \left( \frac{x}{2} \right)^2}  
 
\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)} \\
 
& = \frac{3}{4} \sqrt{x^2 \left( 4 - x^2 \right)} \\
 
& \leq \frac{3}{4} \frac{x^2 + \left( 4 - x^2 \right)}{2} \\
 
& \leq \frac{3}{4} \frac{x^2 + \left( 4 - x^2 \right)}{2} \\
& = \boxed{\textbf{(D) <math>\frac{3}{2}</math>}} ,
+
& = \boxed{\textbf{(D) } \frac{3}{2}} ,
 
\end{align*}
 
\end{align*}
<math></math>
+
</cmath>
  
 
where the inequality follows from the AM-GM inequality and it is binding if and only if <math>x^2 = 4 - x^2</math>.
 
where the inequality follows from the AM-GM inequality and it is binding if and only if <math>x^2 = 4 - x^2</math>.
  
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Revision as of 17:30, 15 November 2023

Solution

Denote by $x$ the length of the shorten base. Thus, the height of the trapezoid is \begin{align*} \sqrt{1^2 - \left( \frac{x}{2} \right)^2} . \end{align*}

Thus, the area of the trapezoid is \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) } \frac{3}{2}} , \end{align*}

where the inequality follows from the AM-GM inequality and it is binding if and only if $x^2 = 4 - x^2$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)