An isosceles triangle has equal sides 4 cm long and the included angle ?. If $\frac { d \theta } { d t } = 2 \mathrm { rad } / \mathrm { min }$ when $\theta = \frac { \pi } { 3 },$ then the rate of change of the area in cm²/min is

Question

An isosceles triangle has equal sides 4 cm long and the included angle ?. If

\frac { d \theta } { d t } = 2 \mathrm { rad } / \mathrm { min }

when

\theta = \frac { \pi } { 3 },

then the rate of change of the area in cm²/min is

Quizplus · Accepted Answer

The area $A$ of an isosceles triangle with two equal sides of length $s$ and included angle $\theta$ is given by the formula $A = \frac{1}{2}s^2\sin\theta$. Differentiating both sides with respect to time $t$, we get $\frac{dA}{dt} = \frac{1}{2} \cdot 2s \cdot \sin\theta \cdot \frac{ds}{dt} + \frac{1}{2}s^2 \cdot \cos\theta \cdot \frac{d\theta}{dt}$. Since $s$ is constant, $\frac{ds}{dt} = 0$, simplifying the expression to $\frac{dA}{dt} = \frac{1}{2}s^2 \cdot \cos\theta \cdot \frac{d\theta}{dt}$.Given $s = 4$ cm, $\theta = \frac{\pi}{3}$, and $\frac{d\theta}{dt} = 2$ rad/min, we substitute these values into the simplified expression:$\frac{dA}{dt} = \frac{1}{2} \cdot 4^2 \cdot \cos\left(\frac{\pi}{3}\right) \cdot 2 = 8 \cdot \frac{1}{2} \cdot 2 = 8$.Since the area is decreasing, the rate of change of the area is $-8$ cm²/min.

An Isosceles Triangle Has Equal Sides 4 Cm Long and the Included