a pool in the shape of a circle measures 10 feet across. One cubic yard of concrete is used to create a circular border of uniform width around the pool. If the border is to have a depth of 3 inches, how wide will the border be (1 cubic yard=27 cubic feet)
area of border= (outerradius^2-innerradius^2) * PI
27= 1/4 ft*area border
solve for outerradius.
To find out the width of the border, we need to calculate the volume of the concrete used for the border first.
The pool in the shape of a circle has a diameter of 10 feet, so its radius is half of that, which is 5 feet.
The border has a depth of 3 inches, which is equivalent to 3/12 = 0.25 feet.
Next, we need to calculate the area of the border.
The area of the border can be found by subtracting the area of the pool from the area of the pool plus the border.
The area of the pool is calculated using the formula: π * r^2, where π is approximately 3.14159 and r is the radius of the pool.
Area of the pool = 3.14159 * (5^2) = 3.14159 * 25 = 78.53975 square feet.
To find the area of the border, we need to add the width of the border to the radius of the pool and then calculate the area using the same formula.
Let's assume the width of the border is w feet.
Area of the border = 3.14159 * (5 + w)^2.
The total volume of the concrete used for the border can be calculated by multiplying the area of the border by its depth:
Volume of the border = Area of the border * Depth = 3.14159 * (5 + w)^2 * 0.25 cubic feet.
Since 1 cubic yard is equal to 27 cubic feet, we can set up the following equation:
(3.14159 * (5 + w)^2 * 0.25) / 27 = 1.
We can solve this equation to find the width of the border.
To find the width of the border, we need to calculate the volume of the border and then divide it by the area of the border's cross-section.
1. Let's start by finding the volume of the border. Since we are given that 1 cubic yard of concrete is used, we can convert it to cubic feet since the pool measurements are in feet. Since 1 cubic yard is equal to 27 cubic feet, the volume of the border would be 27 cubic feet.
2. Next, we need to find the area of the border's cross-section. The border's cross-section is a donut shape formed by the inner pool and the outer border. The area of the cross-section can be calculated by subtracting the area of the pool from the area of the whole border.
- Area of the pool: The pool is a circle with a diameter of 10 feet. The radius of the pool is half of the diameter, so the radius is 10 feet / 2 = 5 feet. The area of the pool is π * r^2, where π is approximately 3.14. Therefore, the area of the pool is 3.14 * (5 feet)^2 = 3.14 * 25 square feet = 78.5 square feet.
- Area of the whole border: The whole border is also a circle with a diameter of (10 feet + 2 * width of the border). Since the width of the border is not known yet, we can use a variable, let's say "w," to represent the width of the border. So, the diameter of the whole border is (10 feet + 2w) feet. The radius of the whole border is half of the diameter, so the radius is (10 + 2w) feet / 2 = (5 + w) feet. The area of the whole border is π * r^2, which is 3.14 * ((5 + w) feet)^2.
- Area of the border's cross-section: Subtract the area of the pool from the area of the whole border: 3.14 * ((5 + w) feet)^2 - 78.5 square feet.
3. Now, we can determine the width of the border by dividing the volume of the border (27 cubic feet) by the area of the border's cross-section.
- Width of the border = Volume of the border / Area of the border's cross-section
- Width of the border = 27 cubic feet / (3.14 * ((5 + w) feet)^2 - 78.5 square feet)
By using this equation, you can determine the width of the border.