Question

If the rate of change of area of rhombus with respect to its side is equal to the side of the rhombus, then the angles of rhombus are

• A. $\frac{\pi }{3}$ and $\frac{2\pi }{3}$
• B. $\frac{\pi }{4}$ and $\frac{3\pi }{4}$
• C. $\frac{\pi }{6}$ and $\frac{5\pi }{6}$
• D. $\frac{5\pi }{12}$ and $\frac{7\pi }{12}$

Answer 1

The correct option is C $\frac{\pi }{6}$ and $\frac{5\pi }{6}$

Let $x$ be the side of the rhombus.
Let $\theta$ be an angle of the rhombus.
Then, area of rhombus is given by $A=x^2\sin \theta$
$\Rightarrow \frac{dA}{dx}=2x\sin \theta$
But, $\frac{dA}{dx}=x$
$\therefore 2x\sin \theta =x\Rightarrow \sin \theta =\frac{1}{2}$
$\Rightarrow \theta =\frac{\pi }{6},\frac{5\pi }{6}$