Question 410522
The smallest area is equal to 1013.53155 as shown in the graph below:


{{{graph(400,400,-24,24,-100,2000,7.141592654*x^2 - 192*x + 2304,1013.53155)}}}


the full picture of this graph is shown below:


{{{graph(400,800,-24,24,-1000,15000,7.141592654*x^2 - 192*x + 2304,1013.53155)}}}


the solution was derived as follows:


let x = length of string that forms the square.
let 24-x = length of string that forms the circle.


circumference of a square is 4*s where s is equal to 1 side of the square.


since the length of the string forms the circumference of the square, we get:


4*s = x which allows us to derive:


s = (x/4)


area of a square is equal to s^2 which is therefore equal to (x/4)^2 which is therefore equal to x^2/16


area of the square is equal to x^2/16


the circumference of a circle is equal to 2*pi*r


since the length  of the remaining string is equal to the circumference of the circle, then we get:


2*pi*r = 24-x


solve for r to get:


r = (24-x) / (2*pi)


the area of a circle is equal to pi * r^2


substitute to get:


area of a circle is equal to pi * ((24-x)^2/(2*pi))^2


simplify this to get:


area of a circle is equal to pi * ((24-x)^2/(4*pi^2)


simplify this further to get:


area of a circle is equal to (24-x)^2 / (4*pi)


simplify this further to get:


area of a circle is equal to (x^2 - 48x + 576) / (4*pi)


let the sum of the area of the square and the circle equal to y.


you get:


y = x^2/16 + ((x^2 - 48x + 576) / (4*pi))


simplify this by finding a common denominator to get:


y = (4*pi*x^2 + 16 * (x^2 - 48*x + 576)) / (4*16*pi)


simplify this further to get:


y = (4*pi*x^2 + 16*x^2 - 768*x + 9216) / (64*pi)


set y = 0 to get:


(4*pi*x^2 + 16*x^2 - 768*x + 9216) / (64*pi) = 0


combine like terms to get:


(4*pi + 16)*x^2 - 768*x + 9216) / 64*pi) = 0


this is your quadratic equation in standard form.


multiply both sides of this equation by (64*pi) to get:


(4*pi + 16)*x^2 - 768*x + 9216) = 0


divide both sides of this equation by 4 to get:


(pi+4)*x^2 - 192*x + 2304 = 0


replace pi with it's constant value of 3.141592654 and combine like terms to get:


7.141592654*x^2 - 192*x + 2304 = 0


in this quadratic equation:


a = 7.141592654
b = -192
c = 2304


max/min point is found by using the formula of x = -b/2a


x = -b/2a becomes:


x = 192/14.28318531 which becomes:


x = 13.44237968


substitute for x in the equation of:


(4*pi + 16)*x^2 - 768*x + 9216) to get:


when x = 13.44237968, y equals 1013.53155


that's the minimum point of your quadratic equation which is modelling the sum of the areas of the square and the circle.