Question

If two sides of a triangle are given, then the area of the triangle will be maximum if the angle between the given sides be.

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{6}$
• D. $\frac{\mathrm{\pi }}{2}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct answer:

Find the angle between the given sides:

Let $a$ and $b$ be given sides, then the area,

$\begin{array}{rcl}A& =& \left(\frac{1}{2}\right)ab\sin C\end{array}$

Differentiate the above area with respect to $C$.

$\begin{array}{rcl}\frac{dA}{dC}& =& \left(\frac{1}{2}\right)ab\frac{d\left(\sin C\right)}{dC}\\ & =& \left(\frac{1}{2}\right)ab\cos C\left[\because \frac{d\left(sinx\right)}{dx}=cosx\right]\end{array}$

$A$ is the maximum, when $\frac{dA}{dC}=0$

$\begin{array}{rcl}\left(\frac{1}{2}\right)ab\cos C& =& 0\\ \cos C& =& 0\\ C& =& {90}^{°}\left[\because cos90°=0\right]\\ & =& \frac{\mathrm{\pi }}{2}\end{array}$

Hence, the correct option is D