The rate of change in area of a triangle having sides 10 cm and 12 cm when the variable angle between them is \(\theta = 60^\circ\), is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 30 cm2/radian
The area of a triangle with two sides a, b and included angle θ is \(\frac{1}{2}ab\sin\theta\):
\( \displaystyle A = \frac{1}{2} \times 10 \times 12 \times \sin\theta = 60\sin\theta \)
The rate of change of the area with respect to θ is
\( \displaystyle \frac{dA}{d\theta} = 60\cos\theta \)
At \(\theta = 60^\circ\):
\( \displaystyle \frac{dA}{d\theta} = 60 \times \frac{1}{2} = 30 \)
So the area changes at 30 cm2 per radian.