Jeff W. answered • 05/31/19
Professional Actuary with Math and Science background
Hi Felix.
This problem requires two main steps and 4 equations to answer. Step 1 is to create a relationship between the circumference of the circle and the circumference of the square in terms of the length of one side of the square (represented by s) knowing that the two added together must equal 8 feet:
The circumference of a circle is 2 times pi times the radius.
The circumference of a square is the sum of all 4 sides, so, s + s + s + s = 4 * s
Circumference of circle + circumference of square = 8
2 * pi * r + 4 * s = 8
2 * pi * r = 8 - 4 * s
r = (8 - 4s) / (2pi)
r = (4 - 2s) / pi
The second part of the problem is to find the formula asked for in the question: "a function representing the area of both square and circle as a function of the length of one side of the square"
The area of a circle is pi * r^2
The area of a square is length * width = s * s = s^2
So the area of the circle plus the area of the square is:
= pi * r^2 + s^2
using the formula we developed in step one, we substitute r for the equation in terms of s
= pi * [ (4 - 2s) / pi]^2 + s^2
= pi * (4-2s)^2 / pi^2 + s^2
= [(4 - 2s) * (4 - 2s)] / pi + s^2
= (16 - 8s - 8s + 4s^2) / pi + s^2
= (16 - 16s + 4s^2) / pi + s^2
The formula can be simplified from here. I don't typically replace pi with an actual number, so my simplified equation would be:
= 16/pi - (16/pi)s + (4/pi)s^2 + s^2
= 16/pi - (16/pi)s +(4/pi)s^2 +(pi/pi)s^2
= 16/pi - (16/pi)s + [(4+pi)/pi]s^2
If pi is substituted in:
= 5.093 - 5.093s + 2.273s^2
You can also note that in the context of this question, this function is only valid from s = 0 (i.e. the rope is not cut and all 8 feet are used for the circle) to s = 2 (the rope is not cut and all 8 feet are used for the square). With those bounds on s, the sum of the areas of the circle and square are limited from 4 to 16/pi (5.093).
