Question 905879: A piece of rope that is 22 feet 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 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 circumference of the circle and Pis the perimeter of the square, then C+P=22.
Enter the expression in simplest form.
f(s)=
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the rope is 22 feet long.
C = circumference of th4e circle = 2*pi*r
r is the radius of the circle.
P = perimeter of the square = 4*s
s is one side of the square.
A = area of the circle = pi*r^2
the formulas you have to work with are:
C = 2*pi*r
P = 4*s
A = pi*r^2
you know that C + P = 22.
that is given.
you can solve for P in this equation to get P = 22 - C
You can solve for A in terms of C as follows:
C = 2*pi*r
A = pi*r^2
If you divide C by 2 and multiply it by r, you get:
A = C*r/2
since C = 2*pi*r, this equation becomes A = 2*pi*r*r/2 which simplifies to A = pi*r^2
so you can use A = C*r/2 in place of A = pi*r^2 because they're equivalent.
you know that C + P = 22
you can solve for C in terms of P to get:
C = 22 - P
you know that P = 4s
you can replace P in the equation of C = 22 - P to get C = 22 - 4s
since C = 2*pi*r, you can replace C in the equation of C = 22 - 4s to get 2*pi*r = 22 - 4s.
you can solve for r in this equation to get r = (22-4s)/(2*pi)
now you're ready to tackle the main equation.
start with A = pi*r^2
replace pi*r^2 with C*r/2 to get A = C*r/2
replace C with 22 - 4s to get A = (22-4s)*r/2
replace r with (22-4s)/(2*pi) to get A = (22-4s) * ((22-4s)/(2*pi)/2
simplify to get A = (22-4s) * (22-4s) / (4*pi)
simplify further to get A = (22-4s)^2 / 4*pi)
simplify further to get A = (484 - 176s + 16s^2) / (4*pi)
simplify further to get A = (121 - 44s + 4s^2) / pi
you can confirm by finding tghe area of the circle in the normal way of A = pi*r^2 and then comparing that result to the area of the circle found by way of A = (121 - 44s + 4s^2) / pi.
The area should be the same.
I confirmed as follows:
let s = 3
you get P = 4s = 12
since C = 22 - P, then C = 10
C = 2*pi*r becomes 10 = 2*pi*r
solve for r to get r = C/(2*pi) which becomes r = 10/(2*pi)
A is equal to pi*r^2 which is equal to pi * (10/(2*pi))^2 which becomes A = 7.957747155...
when s = 3, the formula of A = (121 - 44s + 4s^2)/pi becomes A = (121 - 44*4 + 16*9)/pi which becomes A = (121 - 132 + 36)/pi which becomes A = 25/pi which becomes A = 7.957747155...
The areas are the same so the equations must be good.
are of the circle is equal to pi * r^2
area of the circle in terms of s is equal to (121 - 44s + 4s^2)/pi
|
|
|
