Answer: x = 7.14ft
Explanation:
First, the circle will have a perimeter P = x.
We know that the perimeter of a circle is:
P = 2*pi*r
where pi = 3.14 and r = radius of the circle.
Then we will have:
x = 2*pi*r
r = (x/(2*3.14)) = x/6.28
And the area of a circle of radius r is:
A = pi*r^2 = 3.14*(x/6.28)^2
Now, for the square, the perimeter will be:
P = 16ft - x
And we know that the perimeter of a square of sidelengt s is:
P = 4*s
then:
16ft - x = 4*s
s = (16ft - x)/4 = 4ft - x/4
And the area of a square is:
A = s^2 = (4ft - x/4)*(4ft - x/4).
Now, the total area of the circle + square will be:
Total area = (4ft - x/4)*(4ft - x/4) + 3.14*(x/6.28)^2
We want to find the value of x that minimizes this.
First, let's rewrite the area equation as a quadratic equation:
Ta(x) = (4ft - x/4)*(4ft - x/4) + 3.14*(x/6.28)^2
= 16ft + x^2/16 -2ft*x + 0.08*x^2
= (1/16 + 0.08)*x^2 - 2ft*x + 16ft^2
= 0.14*x^2 - 2ft*x + 16ft^2
This is a quadratic equation with a positive leading coefficient, this means that the arms of the graph will open upwards, then the minimum will be at the vertex of the equation.
To find the vertex we can just take the first derivative, and find at what value of x is equal to zero.
Ta´(x) = 2*0.14*x - 2ft
Ta´(x) = 0.28*x - 2ft
Let´s find x such that this is equal to zero:
Ta´(x) = 0 = 0.28*x - 2ft
x = 2ft/0.28 = 7.14ft
Then x = 7.14ft minimizes the area of the sum of the areas of the circle and square.