The rate of change in area of a triangle having sides 10cm and 12cm when the variable angle between them at an angle \( 60^\circ \), is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 30 \( \mathrm{cm}^2 \) / radian
With two sides and the included angle θ, the area of the triangle is
\( \displaystyle A=\frac{1}{2}(10)(12)\sin\theta=60\sin\theta \)
The rate of change of area with respect to the angle is
\( \displaystyle \frac{dA}{d\theta}=60\cos\theta \)
At \( \theta=60^\circ \):
\( \displaystyle \frac{dA}{d\theta}=60\times\frac{1}{2}=30\ \mathrm{cm}^2/\text{radian} \)
Why the other options are incorrect
Option (2) 120 \( \mathrm{cm}^2 \) / radian — Incorrect. The rate is \( 60\cos 60^\circ=30 \), not 120.
Option (3) \( 30\sqrt{3} \) \( \mathrm{cm}^2 \) / radian — Incorrect. \( 30\sqrt{3} \) is the area itself, \( 60\sin 60^\circ \), not its rate of change.
Option (4) \( 60\sqrt{3} \) \( \mathrm{cm}^2 \) / radian — Incorrect. \( 60\sqrt{3} \) is twice the area \( 30\sqrt{3} \), not the rate of change.