biven that $$ heta $$ is increasing at a rate of 0.05 radians per second, find the rate change of $$ A $$ when $$ heta=\frac{\pi}{6} $$ radians.

Question

biven that $$ 	heta $$ is increasing at a rate of 0.05 radians per second, find the rate change of $$ A $$ when $$ 	heta=\frac{\pi}{6} $$ radians.

UpStudy AI · Accepted Answer

Assume that the area is given by

$$
A=\frac{1}{2}\theta^2.
$$

Then we have a relation between $$ A $$ and $$\theta$$. To find the rate of change of $$ A $$ with respect to time, we differentiate with respect to $$ t $$ using the chain rule:

$$
\frac{dA}{dt}=\frac{dA}{d\theta}\frac{d\theta}{dt}.
$$

Step 1. Differentiate $$ A $$ with respect to $$ \theta $$:

$$
\frac{dA}{d\theta}=\frac{d}{d\theta}\left(\frac{1}{2}\theta^2\right)=\theta.
$$

Step 2. We are given that

$$
\frac{d\theta}{dt}=0.05\quad \text{(radians per second)}.
$$

Step 3. Substitute into the chain‐rule formula:

$$
\frac{dA}{dt}=\theta\left(0.05\right).
$$

Step 4. Evaluate at $$\theta=\frac{\pi}{6}$$:

$$
\frac{dA}{dt}=\left(\frac{\pi}{6}\right)(0.05)=\frac{\pi}{120}.
$$

Thus, the rate of change of $$ A $$ when $$\theta=\frac{\pi}{6}$$ radians is

$$
\frac{dA}{dt}=\frac{\pi}{120}\quad \text{(square units per second)}.
$$
The rate of change of $$ A $$ when $$ \theta = \frac{\pi}{6} $$ radians is $$ \frac{\pi}{120} $$ square units per second.

biven that is increasing at a rate of 0.05 radians per second, find the rate
change of when radians.

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biven that is increasing at a rate of 0.05 radians per second, find the rate change of when radians.