How do you solve this Word Problem

Question

A given length of a string can be made into either the Circumference of a circle with the perimeter of a square. If the area of the circle is 10 in.² more than the area of the square how long is the string?

Answer 1

The circle and square have the same perimeter.

x is the measure of one side of the square.

The perimeter is 4x and the area is x^2

The perimeter of the circle is also 4x = pi * diameter = 2 * pi * radius

2x = pi * radius

x = pi/2 * radius

2x/pi = radius

The area of the circle is 10 square inches more than the area of the square

pi *radius^2 = pi * (2x/pi)^2 = pi ( 4x^2)/ pi^2 = 4x^2/pi

4x^2/pi = x^2 + 10

4x^2 = pix^2 + 10*pi

(4-pi)x^2 = 10*pi

x^2 = 10*pi / (4 - pi)

x = sqrt( 10*pi/ (4-pi)) which is approximately 6.0496217785292062789604541626998...

The string is 4 times that, so

4x = 4 * sqrt( 10*pi/ (4-pi)) = 24.1984871141168251158...