volume of stuff is 2 inches times (area of outer circle - area of inner circle)
22 inches = 1/6 foot
1 yard^3 = 27 ft^3
so
27 = (1/6)(pi)(R^2 - 5^2)
solve that for R
then width of path = R-5 in feet
22 inches = 1/6 foot
1 yard^3 = 27 ft^3
so
27 = (1/6)(pi)(R^2 - 5^2)
solve that for R
then width of path = R-5 in feet
First, let's find the radius of the pool:
The diameter of the pool is given as 10 ft, so the radius is half of that, which is 10 ft / 2 = 5 ft.
Now, let's find the area of the pool:
The area of a circle is calculated using the formula: A = πr^2, where A is the area and r is the radius.
So, the area of the pool is A = π * (5 ft)^2 = 25π square feet.
Next, let's find the radius of the larger circle (pool + border):
Since we want to find the width of the border, let's assume that the radius of the larger circle is (5 ft + x), where x is the width of the border.
Now, let's calculate the area of the larger circle (pool + border):
A = π * (5 ft + x)^2 = 25π + 10πx + πx^2 square feet.
We know that the volume of 1 cubic yard is equal to 27 cubic feet.
Now, let's find the volume of the larger circle:
The volume of a cylinder is calculated using the formula: V = Ah, where V is the volume, A is the base area, and h is the height.
So, the volume of the larger circle is V = (25π + 10πx + πx^2) * (2 inches) = (25π + 10πx + πx^2) * (2/12) feet = (25π + 10πx + πx^2) * (1/6) ft^3.
The volume of the concrete used for the border is the difference between the volume of the larger circle and the volume of the smaller circle:
(25π + 10πx + πx^2) * (1/6) - 25π = 1/3 ft^3.
Now, let's solve the equation to find the value of x, which is the width of the border:
(25π + 10πx + πx^2) * (1/6) - 25π = 1/3
(25π + 10πx + πx^2)/6 - 25π = 1/3
(25π + 10πx + πx^2)/6 = 1/3 + 25π
(25π + 10πx + πx^2)/6 = (1 + 75π)/3
25π + 10πx + πx^2 = 2(1 + 75π)
25π + 10πx + πx^2 = 2 + 150π
Ï€x^2 + 10Ï€x + 25Ï€ - 150Ï€ - 2 = 0
Ï€x^2 + 10Ï€x - 125Ï€ - 2 = 0
Now, we can solve this quadratic equation using the quadratic formula or any other appropriate method to find the value of x, which gives us the width of the border.