a circular pool measures 12 feet across. one cubic yard of concrete is to be used to create a circular border of uniform width around the pool. if the border is to have a depth of 3inches, how wide will the border be? (1cubic yard=27cubic feet)

To find the width of the border, we need to calculate the area of the pool and the area of the border.

1. Let's first calculate the area of the pool:
The pool is circular, and its diameter is given as 12 feet. We can find the radius by dividing the diameter by 2:
Radius of the pool = 12 feet / 2 = 6 feet

The area of a circle is given by the formula: Area = π * (radius)^2
Area of the pool = π * (6 feet)^2

2. Now, let's calculate the area of the border:
To determine the width of the border, we need to subtract the area of the pool from the total area including the border.

One cubic yard of concrete is equal to 27 cubic feet. Since the border has a depth of 3 inches, we need to convert this into feet by dividing by 12:
Depth of the border = 3 inches / 12 = 0.25 feet

The total area including the border is the area of the pool plus the area of the border. The outer radius of the border will be the sum of the radius of the pool and the width of the border:
Outer radius of the border = 6 feet + width of the border

The area of the border is given by: Area = π * (outer radius)^2 - π * (inner radius)^2
Area of the border = π * (6 feet + width of the border)^2 - π * (6 feet)^2

Since the volume of concrete is equal to the area multiplied by the depth, we can set up the equation:
27 cubic feet = (Area of the border) * Depth of the border
27 = [π * (6 + width of the border)^2 - π * 6^2] * 0.25

3. Now, let's solve for the width of the border:
Divide both sides of the equation by 0.25:
108 = π * [(6 + width of the border)^2 - 6^2]

Expand and simplify:
108 = π * [(width of the border)^2 + 12 * width of the border]

Divide both sides of the equation by π:
108 / π = (width of the border)^2 + 12 * width of the border

Rearrange the equation:
(width of the border)^2 + 12 * width of the border - 108 / π = 0

Now, we can solve this quadratic equation using the quadratic formula:

width of the border = [-12 ± √(12^2 - 4 * 1 * (-108 / π)) ] / (2 * 1)

Simplifying further, we have:
width of the border = [-12 ± √(144 + (432 / π))] / 2

Calculating the value within the square root:
value = 144 + (432 / π)

Substituting the value into the equation:
width of the border = [-12 ± √(value)] / 2

You can use a calculator to find the approximate value of the square root and calculate the two possible widths for the border.

To find the width of the circular border, we need to do the following steps:

1. Find the volume of the border:
- Depth of the border = 3 inches = 3/12 feet = 0.25 feet (since 12 inches make a foot).
- The outer radius of the border (including the pool) = (12 feet + 2w), where w is the width of the border.
- The inner radius of the border (excluding the pool) = 12 feet.
- The volume of the border can be calculated as follows:
- Volume of the border = π * (outer radius squared - inner radius squared) * depth of the border.

2. Convert the volume from cubic feet to cubic yards:
- Given that 1 cubic yard = 27 cubic feet, divide the volume by 27 to convert it into cubic yards.

3. Set up and solve the equation to find the width of the border:
- Equate the volume of the border (in cubic yards) to 1 cubic yard and solve for the width (w).

Let's solve it step by step.

Step 1: Find the volume of the border:

Outer radius of the border = 12 feet + 2w
Inner radius of the border = 12 feet
Depth of the border = 0.25 feet

Volume of the border = π * (outer radius squared - inner radius squared) * depth of the border
Volume of the border = π * ((12 + 2w)^2 - 12^2) * 0.25

Step 2: Convert the volume from cubic feet to cubic yards:

Volume of the border (in cubic yards) = Volume of the border / 27

Step 3: Solve for the width of the border:

Volume of the border (in cubic yards) = 1 cubic yard
(π * ((12 + 2w)^2 - 12^2) * 0.25) / 27 = 1

Now, you can solve the equation to find the value of w, which will give you the width of the border.

If the width is w inches, the volume of the border (in^3) is

pi((72+w)^2 - 72^2)*3 = 46656
w = 28.7