Cody has a circular fish pond with a diameter of 6.4 feet around which he wants to put a sidewalk. He has enough concrete left over to cover an area of 65 square feet. Assuming the sidewalk is of uniform width, how wide can the sidewalk around the pond be?

As = pi*r^2 = 65 Ft^2.=Area of sidewalk.

r^2 = 20.69
r2 = 4.55 Ft. = Radius of sidewalk.

r1 = 6.4/2 = 3.2 Ft. = radius of pond.

W = r2 - r1 = 4.55 - 3.2 = 1.35 Ft. =
Width of sidewalk.

To find the width of the sidewalk around the pond, we need to subtract the area of the pond from the total area covered by the sidewalk and the pond.

First, let's find the area of the pond. The pond is circular, so we can use the formula for the area of a circle:

Area = π * radius^2

Since the diameter of the pond is given as 6.4 feet, the radius is half the diameter, so the radius of the pond is 6.4 / 2 = 3.2 feet.

Now, calculate the area of the pond:

Area of the pond = π * (3.2)^2 = 32.16 square feet (approximately)

Next, we subtract the area of the pond from the total area covered by the sidewalk and the pond:

Area of the sidewalk + pond - Area of the pond = Concrete area

Let's denote the width of the sidewalk as 'w'. The width of the sidewalk will be the same both on the inside and outside of the pond. Therefore, the diameter of the pond plus twice the width of the sidewalk will give us the diameter of the whole area covered by the sidewalk and the pond.

Diameter of the sidewalk + pond = Diameter of the pond + 2 * width of sidewalk
=> Diameter of the sidewalk + pond = 6.4 + 2w

Since the diameter is related to the radius by the formula: diameter = 2 * radius, the above equation becomes:

2 * (radius of the sidewalk + pond) = 6.4 + 2w

Now, we can calculate the radius of the sidewalk and pond:

Radius of the sidewalk + pond = (6.4 + 2w) / 2 = 3.2 + w

Using the formula for the area of a circle, we can calculate the total area covered by the sidewalk and the pond:

Area of the sidewalk + pond = π * (3.2 + w)^2

Now, we subtract the area of the pond from the total area covered by the sidewalk and the pond:

π * (3.2 + w)^2 - 32.16 = 65

Solving this equation will give us the width (w) of the sidewalk:

π * (3.2 + w)^2 = 97.16

(3.2 + w)^2 = 97.16 / π

Taking the square root of both sides:

3.2 + w = √(97.16 / π)

w = √(97.16 / π) - 3.2

Using a calculator or approximating π to 3.14, we can determine the width of the sidewalk.

w ≈ √(97.16 / 3.14) - 3.2

w ≈ √(30.94) - 3.2

w ≈ 5.56 - 3.2

w ≈ 2.36

Therefore, the width of the sidewalk around the pond can be approximately 2.36 feet.