SOLUTION: A wire 360 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 pie

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Question 104247This question is from textbook precalculus fith edition mathematics for calculus
: A wire 360 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? This question is from textbook precalculus fith edition mathematics for calculus

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
For both figures, you have to find out the relationship between its perimeter (the length of wire) and the area.
For a square of side L. The perimeter is 4L. The area is L*L.
P=4L
L=P%2F4
A=L%2AL
A=P%5E2%2F16
For a circle of radius, R, the perimeter is 2*pi*R and the area is pi*R*R.
P=2%28pi%29R
R=P%2F%282%28pi%29%29
A=%28pi%29R%5E2
A=%28pi%29%28P%2F%282%28pi%29%29%5E2%29
A=P%5E2%2F%284%28pi%29%29
The perimeter of the square is the length of wire used in the square.
P%5B1%5D
The perimeter of the circle is the length of wire used in the circle.
P%5B2%5D
The areas of the square and circle equal each other.
1.P%5B1%5D%5E2%2F16=P%5B2%5D%5E2%2F%284%28pi%29%29
and the length of the two cut wires equals 360 inches.
2.P%5B1%5D%2BP%5B2%5D=360
From equation 1,
P%5B1%5D%5E2%2F16=P%5B2%5D%5E2%2F%284%28pi%29%29
P%5B1%5D%5E2=16%2AP%5B2%5D%5E2%2F%284%28pi%29%29
P%5B1%5D=2%2AP%5B2%5D%2Fsqrt%28%28pi%29%29
Substitute into equation 2,
P%5B1%5D%2BP%5B2%5D=360
2%2AP%5B2%5D%2Fsqrt%28%28pi%29%29%2BP%5B2%5D=360
P%5B2%5D%281%2B2%2Fsqrt%28%28pi%29%29%29=360
P%5B2%5D=360%2F%281%2B2%2Fsqrt%28%28pi%29%29%29
P%5B2%5D=169.1 inches
From equation 2,
P%5B1%5D%2BP%5B2%5D=360
P%5B1%5D=360-P%5B2%5D
P%5B1%5D=190.9 inches
Check your answer
The area of the square is
A=P%5E2%2F16
A=2278
The area of the square is
A=P%5E2%2F%284%28pi%29%29
A=P%5E2%2F%284%28pi%29%29
A=2276
The answers are off a little from roundoff error but good enough. Good answer.