Area and limit of the shaded region...?

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September 15th 2008, 08:21 PM
damselfly

Area and limit of the shaded region...?

I'm not sure if I did this is right, but here's the question:
http://i3.photobucket.com/albums/y62...calculus-1.jpg
Consider the shaded region outside the sector of a circle of radius 10 meters and inside a right triangle.

a) write the area A= f(θ) of the region as a function of θ.

My equation: f(θ) = 50θ - 50 arctanθ

b) What is the domain of the function?

All real numbers.

c) Find lim (θ--> π/2-) A.

Infinity... why?

Thanks!
September 16th 2008, 10:22 AM
Laurent

Quote:

Originally Posted by damselfly

a) write the area A= f(θ) of the region as a function of θ.

My equation: f(θ) = 50θ - 50 arctanθ

Not quite. The area of the triangle is $50\tan\theta$ , from which you substract the area of the piece of disc ( $50\theta$ , indeed), so that $A=50(\tan\theta-\theta)$ .

Now you can reconsider the other questions.

Laurent.
September 16th 2008, 10:24 AM
Soroban

Hello, damselfly!

Quote:

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 $A$ of the region as a function of $\theta.$

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

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

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

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

Quote:

b) What is the domain of the function?

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

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

Quote:

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

Answer: Infinity . . . why?

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

Code:

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

And the area of the "triangle" becomes infinite.
September 16th 2008, 02:18 PM
damselfly

Thank you so much!