Consider the shaded region outside the sector of a circle of radius 12 meters and inside a right triangle.
a) write the area A= f(θ) of the region as a function of θ.
I found the area of the triangle and then subtracted the area of the sector and got
My equation: $f(θ) = 72 tan \theta – 72\theta = 72(tan \theta – \theta)$ and I know for sure this is correct
b) What is the domain of the function?
I'm not sure how to find this but I'm leaning towards $0 < \theta \leq \frac{\pi}{2}$. My options are
$0 < \theta < \frac{\pi}{2}$
$0 < \theta \leq \frac{\pi}{2}$
$0 \leq \theta \leq \frac{\pi}{2}$
$-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$
c)What does the area approach as $\theta$ approaches $\frac{\pi}{2}$?
I think its $\infty$ but my options are
$-\infty$
$100$
$50$
$\infty$
$0$
$80$
or $40$
My main struggle is with b and c. Can anyone tell me if these answers are correct.
Best Answer
Yes area of $BCD$ is $f(\theta)=\frac12r^2\tan\theta-\frac12r^2\theta=72(\tan\theta-\theta)$
For domain:
For the limit: $$\lim_{\theta\to\pi/2}f(\theta)=72\lim_{\theta\to\pi/2}(\tan\theta-\theta)=\infty$$ As $\lim_{\theta\to\pi/2}\tan\theta=\infty$ and $\lim_{\theta\to\pi/2}\theta=\pi/2$