A piece of wire 360 inches long is cut into two pieces.

Question

A piece of wire 360 inches long is cut into two pieces. One piece is formed into a square and the other is formed into a circle such that the two figures have the same area. To the nearest tenth of an inch, the length of the wire used to form the square is?and the length used to form the circle is? 

Answer 1

The two pieces add up to 360 inches. 

 

 

Let the length of wire that forms a square = x

Let the length of wire that forms a circle = y

 

The lengths of the shapes these wires form is the same as their perimeters. 

 

For square

 

Perimeter = x

Each side = x / 4

Area  = x² / 16

 

 

For circle

 

y = 2πr  ----> perimeter

 

Area = πr²

 

 

Substituting the perimeter into area ,

 

Area = π(y / 2π)²

Area = y² / (4π)

 

 

Set equations that represents the situation.

 

x + y = 360           eq1

 

x² / 16 = y² / (4π)            eq2     (this equation can be rewritten using cross-multiplication)

 

16y² = 4πx²                  eq2

 

 

We have a system of equations. 

 

We can substitute eq1 into eq2 so that we have only one equation with only one variable.  Let's have an equation in terms of x.

 

16(360 - x)² = 4πx²

 

16(360 - x)(360 - x) = 4πx²

 

16(129600 - 720x + x²) = 4πx²

 

 

Okay.  Solve for x from this equation.  Since this equation has a degree of 2, you will obtain two solutions.  Once you have your x values, substitute them into eq1 to solve for y.  As a last step, you will need to verify these solution pairs to make sure they satisfy the initial conditions.