The 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 of the two circles will be covered with grass. What is the area of that section??

I assume you meant

(x+2)^2 + (y-6)^2 = 16 and (x+2)^2 + (y+6)^2= 81
you state that the two circles are concentric, but according to your equations the first one has a centre of (-2,6) and (-2,-6) so there is clearly a typo.

yes all the 2's outside of the parenthesis are squared

Please verify and correct the typo.

The question says the circles are concentric, but the equations give two circles of radii 4 and 9 which intersect each other.
Would the circles be non-concentric, or one of the equations had a typo?

To find the area of the grassy section between the two concentric circles, we need to find the difference between the areas of the larger circle and the smaller circle.

The equation of the larger circle is:

(x + 2)^2 + (y + 6)^2 = 81

The equation of the smaller circle is:

(x + 2)^2 + (y - 6)^2 = 16

Let's simplify these equations to find the radii of the circles.

For the larger circle equation, we have:

(x + 2)^2 + (y + 6)^2 = 81

Expanding this equation:

x^2 + 4x + 4 + y^2 + 12y + 36 = 81

x^2 + y^2 + 4x + 12y - 41 = 0

Rearranging the terms:

x^2 + 4x + y^2 + 12y = 41

Completing the square for x:

x^2 + 4x + 4 + y^2 + 12y = 41 + 4

(x + 2)^2 + y^2 + 12y = 45

(x + 2)^2 + (y + 6)^2 = 45

Comparing this equation with the standard form of a circle equation:

(x - h)^2 + (y - k)^2 = r^2

We can see that the center (-2, -6) and the radius is √45.

Now, let's simplify the equation for the smaller circle:

(x + 2)^2 + (y - 6)^2 = 16

Comparing this equation with the standard form of a circle equation, we can see that the center is (-2, 6) and the radius is √16 = 4.

To find the area of each circle, we can use the formula: A = π * r^2.

The area of the larger circle is:

A_larger = π * (√45)^2

A_larger = 45π

The area of the smaller circle is:

A_smaller = π * (4)^2

A_smaller = 16π

Now, to find the area of the grassy section, we subtract the area of the smaller circle from the area of the larger circle:

A_grassy_section = A_larger - A_smaller

A_grassy_section = 45π - 16π

A_grassy_section = 29π

So, the area of the grassy section between the two concentric circles is 29π.