Word Problems

A piece of rope that is 18 feet long is cut into pieces. One piece is used to form a circle and the
other used to form a square. Write a function f representing the area of the circle as a function
of the length of one side of the square s.


Hint : If C is the diameter of the circle and P is the perimeter of the square, then C + P = 18.


Answer

f(s) = (9-2s)2/π , the area of the circle as a function of the length of one side of the
square.


Explanation

Let
Length of the rope= 18 ft
C = circumference of circle
P = perimeter of square
s = side of the square
r = radius of the circle

Required
f(s) = ? , Area of circle as a function of the length of one side of the square

Solution
C = 2πr , circumference of a circle
P = 4s , perimeter of a square

C+P = 18 ; substitute the formulas
2πr + 4s = 18 ; get the expression for radius,r ;
2πr = 18-4s
r = (18-4s)/2π ; simplifying by dividing by 2
r = (9-2s)/π >>equation 1

Solve for the area of circle as a function of the length of one side of the square, f(s) ;

f(s) = πr2 ; substitute equation 1 to the r
= π((9-2s)/π))2
= π (9-2s)2/π2
= π (81-36s+4s2)/π2
f(s) = (81-36s+4s2)/π
f(s) = (9-2s)2/π , the area of the circle as a function of the length of one side of the
square.