## [ Quadrilaterals (extreme properties).]

For what value of the height does a rectangular trapezoid with an acute angle of $30^{\circ}$ and a perimeter of 6 have the maximum area

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Express the area of the given trapezoid through its height and apply the Cauchy inequality.

## Solution

Let $h$ be the height of the trapezoid, $3 x$ be the sum of the bases. Then the larger lateral side of the trapezoid is $2 h$, and the perimeter is $3 x + 3 h = 6$. The area of the trapezoid is $\frac{3 x h}{2} = \frac{3}{2} \left( \frac{x + h}{2} \right)^{2} = \frac{3}{2}$, and equality is achieved when $h = x = 2 - h$, that is, when $h = 1$.

## Answer

When $h = 1$.