Finding Length and Width

Question

A wire 310 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two figures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?

Answer 1

Let square have side s and circle have radius r.

 

Areas have to be the same, so

 

s² = (pi)r²

 

 

Perimeters have to sum to 310

 

4s + 2(pi)r = 310

 

Solve area relation for either variable, plug into perimeter equation and solve.

 

Solving for s,

 

s = r(sqrt(pi))

 

4r(sqrt(pi)) + 2(pi)r = 310

 

r(4(sqrt(pi)) + 2(pi)) = 310

 

r(13.37) = 310

 

r = 23.2 inches.

 

Go back and solve for s.

 

s = r(sqrt(pi)) = 23.2(sqrt(pi)) = 41.1 inches

 

4s = 164.4 inches.

 

Remainder of wire is for circle.

 

310 - 164.4 = 145.6 inches.

 

Check by calculating circumference of circle of radius 23.2

 

C = 2(pi)r = 2(3.14)(23.2) = 145.7 inches.  Close enough.

 

 

The two pieces of wire are 164.4 inches and 145.6 inches long.