Question

A triangular partk is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is:

• A. $\sqrt{\frac{x^3}{8}}$
• B. $\frac{1}{2}{x}^2$
• C. $\pi x^2$
• D. $\frac{3}{2}{x}^2$

Answer 1

The correct option is B $\frac{1}{2}{x}^2$

Let the angle opposite to the third side be $A$

Area $\Delta =\frac{1}{2}\left(x\right)\left(x\right)\sin A=\frac{1}{2}{x}^2\sin A$

$\Delta$ will be maximum when $\sin A$ is maximum $⟹\sin A=1⟹A={90}^{\circ }$

maximum area enclosed by park is $\frac{1}{2}{x}^2$