Respuesta :
The function g representing the area of the square as a function of the radius of the circle r is given as:
[tex]g(r) = 49 - 22r + \frac{121r^2}{49}[/tex]
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:
[tex]c = 2 \pi r[/tex]
Where, "r" is the radius of circle
Substitute the above circumference in eqn 1
[tex]2 \pi r + p = 28[/tex]
[tex]p = 28 - 2 \pi r[/tex] ----------- 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
[tex]4a = 28 - 2 \pi r\\\\a = \frac{28 - 2 \pi r}{4}\\\\a = 7 - \frac{ \pi r}{2}[/tex]
The area of square is given as:
[tex]area = (side)^2\\\\area = a^2[/tex]
Substitute the value of "a" in above area expression
[tex]area = (7 - \frac{ \pi r}{2})^2[/tex] ------ eqn 3
We know that,
[tex](a - b)^2 = a^2 - 2ab + b^2[/tex]
Therefore eqn 3 becomes,
[tex]area = 7^2 -2(7)(\frac{\pi r}{2}) + (\frac{ \pi r}{2})^2\\\\area = 49 - 7 \pi r + \frac{ (\pi)^2 r^2 }{4}[/tex]
[tex]\text{ substitute } \pi = \frac{22}{7}[/tex]
[tex]area = 49 - 7 \times \frac{22}{7} \times r + (\frac{22}{7})^2 \times \frac{r^2 }{4}\\\\area = 49 - 22r + \frac{121}{49} \times r^2\\\\area = 49 - 22r + \frac{121r^2}{49}[/tex]
Let g(r) represent the area of the square as a function of the radius of the circle r, then we get
[tex]g(r) = 49 - 22r + \frac{121r^2}{49}[/tex]
Thus the function is found
