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 | ||
| − | < | + | <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) | + | & = \boxed{\textbf{(D) } \frac{3}{2}} , |
\end{align*} | \end{align*} | ||
| − | + | </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 18:30, 15 November 2023
Solution
Denote by 
the length of the shorten base.
Thus, the height of the trapezoid is
Thus, the area of the trapezoid is
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)
