Question

the y - coordinate on a circle of radius 7 centered at the origin is given by y = 7sin(θ), where θ is the angle of rotation on the circle measured counterclockwise from the positive x - axis. if θ is changing at a rate of 0.05 radians per second, how is the y - coordinate of the point on the circle changing when θ = 3π/4.

Explanation:

Step1: Differentiate y with respect to θ

We know that $y = 7\sin(\theta)$. Using the derivative formula $\frac{d}{d\theta}\sin(\theta)=\cos(\theta)$, we get $\frac{dy}{d\theta}=7\cos(\theta)$.

Step2: Use the chain - rule

The chain - rule states that $\frac{dy}{dt}=\frac{dy}{d\theta}\cdot\frac{d\theta}{dt}$. We are given that $\frac{d\theta}{dt}=0.05$ radians per second.

Step3: Evaluate at $\theta=\frac{3\pi}{4}$

First, find $\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}$. Then, substitute $\frac{dy}{d\theta}=7\cos(\theta)$ and $\frac{d\theta}{dt}=0.05$ into the chain - rule formula. $\frac{dy}{dt}=7\cos(\theta)\cdot0.05$. Substitute $\theta = \frac{3\pi}{4}$ into the equation: $\frac{dy}{dt}=7\times(-\frac{\sqrt{2}}{2})\times0.05=-\frac{7\sqrt{2}}{40}\approx - 0.247$.

Answer:

$- \frac{7\sqrt{2}}{40}\text{ or approximately }-0.247$