Question

Two sides of a triangle are $8 m$ and $5 m$ in length. The angle between them is increasing at the rate $0.08 \,rad / s .$ 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. $0.4\, m^2/sec$
• B. $0.8 \,m^2$/sec
• C. $0.6\,m^2/sec$
• D. $0.04\,m^2/sec$
• E. $0.08\, md^2/sec$

Answer 1

Answer: (B)

Given, $\Delta A B C$ with $A B=8 m$ and
$A C=5 m$ and $\angle A=\theta=\frac{\pi}{3}$

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