Question

A piece of rope there is 28 feet long is cut into two pieces. One is use to form a circle and others used to form a square. Write a function G representing the area of the square as a function of the radius of the circle R

Answer 1

The function g representing the area of the square as a function of the radius of the circle r is given as:

`g(r) = 49 - 22r + (121r^2)/(49)`

Solution:

Given that,

length of rope = 28 feet

Let "c" be the circumference of circle

Let "p" be the perimeter of square

Therefore,

length of rope = circumference of circle + perimeter of square

c + p = 28 ------- eqn 1

The circumference of circle is given as:

`c = 2 \pi r`

Where, "r" is the radius of circle

Substitute the above circumference in eqn 1

`2 \pi r + p = 28`

`p = 28 - 2 \pi r` ----------- eqn 2

If "a" is the length of each side of square, then the perimeter of sqaure is given as:

p = 4a

Substitute p = 4a in eqn 2

`4a = 28 - 2 \pi r\\\\a = (28 - 2 \pi r)/(4)\\\\a = 7 - ( \pi r)/(2)`

The area of square is given as:

`area = (side)^2\\\\area = a^2`

Substitute the value of "a" in above area expression

`area = (7 - ( \pi r)/(2))^2` ------ eqn 3

We know that,

`(a - b)^2 = a^2 - 2ab + b^2`

Therefore eqn 3 becomes,

`area = 7^2 -2(7)((\pi r)/(2)) + (( \pi r)/(2))^2\\\\area = 49 - 7 \pi r + ( (\pi)^2 r^2 )/(4)`

`\text{ substitute } \pi = (22)/(7)`

`area = 49 - 7 * (22)/(7) * r + ((22)/(7))^2 * (r^2 )/(4)\\\\area = 49 - 22r + (121)/(49) * r^2\\\\area = 49 - 22r + (121r^2)/(49)`

Let g(r) represent the area of the square as a function of the radius of the circle r, then we get

`g(r) = 49 - 22r + (121r^2)/(49)`

Thus the function is found