Question

A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. if the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon

Answer 1

Let a and b be sides of pentagon, a be the side of triangle.

$$S=ab+ \frac {a^2 \sqrt{3}}{4}$$

$$3a+2b=P$$

$$b=0.5P-1.5 a$$

$$S=a(0.5P-1.5a)+ \frac {a^2 \sqrt{3}}{4}$$

$$S^/_a=0.5P-2a- \frac{ \sqrt {3}}{2}a=0$$

$$a=\frac {P}{4- \sqrt{3}}$$

$$b=\frac {0.5-0.5\sqrt{3}}{4-\sqrt{3}}P$$