Garden Area Problem. A designer created a garden

from two concentric circles whose equations are as follows:
(x+2)^2+(y-6)^2=16 and (x+2)^2+(y-6)^2=81
The area between the circles will be covered with grass. What is the area of that section?

How do you do this?

The equation of a standard circle with centre at (a,b) and radius r is

(x-a)² + (y-b)² = r²

By inspection of the given circle,
(x+2)^2+(y-6)^2=16 and (x+2)^2+(y-6)^2=81
we conclude that both have centres at (-2,6), therefore they are concentric.

The radii of the circles are √16=4 and √81=9.

The area between the two circles are therefore
πr1²-πr2²
=π(9²-4²)
=65π

do you have any sites that explain this as well?

The subject should be in any pre-calc textbook. Alternatively, you can google

"equation of a circle".

To find the area between the two concentric circles, you can calculate the areas of the individual circles and then subtract the smaller area from the larger one. Here's how you can do it step by step:

1. Start with the equations of the two concentric circles:
(x+2)^2 + (y-6)^2 = 16 --- Equation of the smaller circle
(x+2)^2 + (y-6)^2 = 81 --- Equation of the larger circle

2. The general form of a circle equation is (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius. By comparing the given equations to the general form, we can determine the centers and radii of the circles:
Center of both circles: (-2, 6)
Radius of the smaller circle: sqrt(16) = 4
Radius of the larger circle: sqrt(81) = 9

3. Calculate the areas of the individual circles using the formula A = πr^2, where A is the area and r is the radius:
Area of the smaller circle: A1 = π * 4^2 = 16π
Area of the larger circle: A2 = π * 9^2 = 81π

4. Finally, subtract the area of the smaller circle from the area of the larger circle to find the area between them:
Area between the circles = A2 - A1 = 81π - 16π = 65π

Therefore, the area between the two concentric circles is 65π. The exact numerical value depends on the precision required in the answer.