write a function g representing the area of the square as a function of the radius of the circle r.

Question

A piece of rope that is 20feet 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.Then write a function grepresenting the area of the square as a function of the radius of the circle r.

Answer 1

The circumference of the circle will be the same as the perimeter of the square, which is 4s.

 

The circumference of the circle is 2πr, so:

 

2πr = 4s

r = 4s / 2π = 2s / π

 

But the area of the circle is πr² = π (2s / π)² = π (4s² / π²) = 4s² / π.   That's the Area of the circle as a function of the length of a side of the square.

 

 

Next, go back to 2πr = 4s, but this time solve for s instead of r:

 

s = 2πr / 4 = πr / 2

 

The area of the square is s² which is equal to (πr / 2)² = π²r² / 4.  That's the Area of the square as a function of the radius of the circle.

 

 

 

 

 

Answer 2

Kelsey,

 

Let s = the length of one side of a square (in feet)

     r = the radius of a circle (in feet)
     c = the circumference of the circle or length of rope the circle requires (in feet)

f(s) = the function that gives the area of a circle in terms of s (in square feet)

g(r) = the function that gives the area of a square in terms of r ( in square feet)

 

Then, the circumference of the circle plus the perimeter of the square must equal total rope length

c + 4s = 20,  so c = 20 - 4s

But c = 2Πr = 20 - 4s 

So  r = (10 - 2s)/Π

And f(s) = area of circle = Π r² =  (10 - 2s)² / Π

 

Also, from  r = (10 - 2s)/Π ,   s = (10 - Π r) / 2

So  g(r) = area of square = s² =  (10 - Π r)² / 4