Hello, damselfly!

Consider the shaded region outside the sector of a circle of radius 10 meters

and inside a right triangle. Code:

B *
|:* D
|:::*
h |:* *
|* *
| θ *
C * - - - - - * A
10

a) Write the area $\displaystyle A$ of the region as a function of $\displaystyle \theta.$

In right triangle $\displaystyle BCA\!:\;\;\tan\theta = \frac{h}{10} \quad\Rightarrow\quad h \:=\:10\tan\theta$

Area of $\displaystyle \Delta BCA \:=\:\frac{1}{2}(10)(10\tan\theta) \:=\:50\tan\theta$

Area of sector $\displaystyle ACD \;=\;\frac{1}{2}(10^2)\theta \;=\;50\,\theta$

Therefore: .$\displaystyle A \;=\;50\tan\theta - 50\,\theta \quad\Rightarrow\quad\boxed{ A \;=\;50(\tan\theta - \theta)}$

b) What is the domain of the function?

$\displaystyle \theta$ could be any acute angle from 0° to *less than* 90°.

. . . $\displaystyle 0 \:\leq \:\theta \:<\:\frac{\pi}{2}$

c) Find: .$\displaystyle \lim_{\theta\to\frac{\pi}{2}^-} A$

Answer: Infinity . . . why?

As $\displaystyle \theta$ increases to $\displaystyle \frac{\pi}{2}$, the radius $\displaystyle AD$ becomes vertical.

Code:

|:::::::::|
|:::::::::|
|:::::::::|
|:::::::* * D
|:::* |
|:* |
|* |
| θ |
C * - - - - * A
10

And the area of the "triangle" becomes infinite.