Answer:
[tex]\displaystyle \frac{dx}{d\theta}=5\text{ ft/rad}[/tex]
Step-by-step explanation:
Please refer to the diagram below.
We can use the sine function. Recall that the sine function is the ratio between the opposite side and the hypotenuse. Hence:
[tex]\displaystyle \sin(\theta)=\frac{x}{10}[/tex]
We want to find how fast x changes with respect to θ. So, we want to find dx/dθ. Therefore, let's take the derivative of both sides with respect to θ:
[tex]\displaystyle \frac{d}{d\theta}[\sin(\theta)]=\frac{d}{d\theta}\left[\frac{1}{10}x\right][/tex]
Differentiate. Since it's with respect to θ, we can differentiate the left-hand side like normal. On the right, we will implicitly differentiate. This yields:
[tex]\displaystyle \cos(\theta)=\frac{1}{10}\frac{dx}{d\theta}[/tex]
Multiply both sides by 10. So, dx/dθ is:
[tex]\displaystyle \frac{dx}{d\theta}=10\cos(\theta)[/tex]
We want to find dx/dθ when θ is π/3. Thus, substitute π/3 for θ:
[tex]\displaystyle \frac{dx}{d\theta}=10\cos\left(\frac{\pi}{3}\right)[/tex]
Evaluate:
[tex]\displaystyle \frac{dx}{d\theta}=10\left(\frac{1}{2}\right)=5\text{ ft/rad}[/tex]
So, x is changing with respect to θ at a rate of 5 feet per radian.
