
Diameter of A Circle
I'm having trouble understanding this problem:
There is a circular town of unknown diameter having four gates located exactly at the north, east, south and west of the town. A tall tree stands 3 li (Note: 1 li = .33 mi) from outside the north gate. When you exit the south gate and turn dues east, you must walk 9 li before you can see the tree.
a. Draw a diagram on your own graph paper to represent this problem. (Hint: You need to set up two right triangles in your diagram to solve this problem.)
I've done this part.
b.Show that the radius, r, of the town must satify the polynomial equation $\displaystyle 4r^4 + 12r^3 + 9r^2  486r 729 = 0$
OK. This part I am confused on...
c. What is the diameter?
I found out the the diameter to be 9 li by using my Calculator and plugging that equation into it. It turns out that when r = 4.5 it satisfies the problem.
My problem is that I don't understand how you get 4.5 li as the radius. And I need to show work for this problem.

What you could do to find that quartic polynomial is to use similar triangles.
$\displaystyle \frac{r}{r+3}=\frac{9}{\sqrt{(2r+3)^{2}+81}}$
Square both sides and bring everything to one side via cross multiplication:
$\displaystyle r^{2}((2r+3)^{2}+81)81(r+3)^{2}=0$
Expand out and get:
$\displaystyle 4r^{4}+12r^{3}+9r^{2}486r729=0$
If we solve this we find it has two real solutions, one is 9/2. That's the one.

Alright thanx alot. I finally understand.