Question

Two sides of a triangle are $8m$ and $5m$ in length. The angle between them is increasing at the rate $0.08rad/s.$ When the angle between the sides of fixed length is $\frac{\pi }{3}$, the rate at which the area of the triangle is increasing, is

• A. $0.4m^2/sec$
• B. $0.8m^2$/sec
• C. $0.6m^2/sec$
• D. $0.04m^2/sec$
• E. $0.08m{d}^2/sec$

Answer 1

Answer: (B)

Given, $\Delta ABC$ with $AB=8m$ and
$AC=5m$ and $\angle A=\theta =\frac{\pi }{3}$

Let $b=5m,c=8m$
Area $=\frac{1}{2}bc\sin A$
$A=\frac{1}{2}\times 5\times 8\sin \theta =20\sin \theta$
On differentiating, we get
$\frac{dA}{dt}=20\cos \theta \cdot \frac{d\theta }{dt}$
$=20\cos \frac{\pi }{3}\times 0.08$
$=0.8m^2/s$