5. A piece of rope is cut into two pieces. One piece is used to form a circle and the other is used to form a square. Use a length of rope of 20 feet to write a function f representing the area of the circle as a function of the length of one side s of the square.
To find the area of the circle formed by one piece of the rope, we need to calculate the radius of the circle. Since the rope was cut into two pieces, we can assume that one piece was used to form the circumference of the circle.
The formula for the circumference of a circle is C = 2πr, where π is a mathematical constant approximately equal to 3.14 and r is the radius. Therefore, we can say that the length of one piece of the rope is equal to the circumference of the circle, which in turn is equal to 2πr.
Let's call the length of the rope used for the circumference Lc. We know that Lc is equal to 20 feet, so we can write:
Lc = 2πr
To solve for the radius, we need to rearrange the equation as follows:
r = Lc / (2π)
Now, we can substitute the value of Lc into the equation:
r = 20 / (2π)
Next, we need to find the area of the circle. The formula for the area of a circle is A = πr^2. Using the value of r we just calculated, we can write the area as:
A = π(20 / (2π))^2
Simplifying the equation further, we have:
A = π(10 / π)^2
A = π(100 / π^2)
A = 100π / π^2
A = 100 / π
Therefore, the area of the circle formed by one piece of the rope can be represented as a function of the length of one side s of the square:
f(s) = 100 / π, where s is the length of one side of the square.