A piece of rope that is 24feet long is cut into two pieces. One piece is used to form a circle and the other used to form a square. Write a function frepresenting the area of the circle as a function of the length of one side of the square s.

Ac = (24- (s / 4)) ^ 2 / (4 pi)

Step-by-step explanation:

Because the piece is rope is divided into two pieces we have to:

c + s = 24, where c is the circumference of circle and s is the perimeter of square. Therefore the circumference of circle would be equal to:

c = 24-s

Let As the area of the square be, we know that the perimeter is the sum of all sides and a square has four sides, therefore the value of each side would be the perimeter divided by four. And the area of the square is a side raised to the square, as follows:

As = (s / 4) ^ 2

Now knowing that the circumference of circle is equal to the radius, by the number pi, by two, we have to:

c = 2 * pi * r

rearranging we have:

r = c / (2 * pi)

Let Ac be the area of the circle we have the radius squared by the number pi, like this:

Ac = pi * r ^ 2

But we already know the equivalence of the radius, replacing:

Ac = pi * (c / (2 * pi)) ^ 2

Now replacing the circumference of circle with its equivalence with the perimeter of the square we have to:

Ac = pi * ((24-s) / (2 * pi)) ^ 2

Now knowing that the one side of the square is equal to a quarter of the perimeter of the square we have:

Ac = (24- (s / 4)) ^ 2 / (4 pi)

That would be the area of the circle as a function of one side of the square.