SOLUTION: 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 pi

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Question 403986: 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 by ankor@dixie-net.com(22740) About Me  (Show Source):
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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)?
:
Let s = the side of the square
then
4s = the perimeter of the square
s^2 = the area of the square
:
Let r = the radius of the circle
then
2%2Api%2Ar = the circumference of the circle
pi%2Ar%5E2 = area of the circle
:
"the two figures have the same area,
s^2 = pi%2Ar%5E2
s = sqrt%28pi%2Ar%5E2%29
:
"A wire 310 in. long is cut into two pieces."
:
Perimeter + Circumference = 310 inches
4s + 2%2Api%2Ar = 310
:
Replace s with sqrt%28pi%2Ar%5E2%29
4%28sqrt%28pi%2Ar%5E2%29%29+%2B+%282%2Api%2Ar%29 = 310
:
simplify, divide by 2, Extract r
2r%28sqrt%28pi%29%29+%2B+%28pi%2Ar%29 = 155
:
Change pi to decimal values
2r(1.772) + r(3.1416) = 155
3.545r + 3.1416r = 155
6.6865r = 155
r = 155%2F6.6865
r = 23.18
:
Find the circumference
C = 2%2Api%2A23.18
C = 145.7 inches the length of the wire to make the circle
then
310 - 145.7 = 164.3 inches is the length to make the square
:
:
Check this by finding the area of circle and the square
Square: (164.3/4)^2 = 1687.155 sq/inches
Circle: pi%2A23.18%5E2 = 1688.0 sq/inches, discrepancy is from rounding off