SOLUTION: You have a wire that is 53 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle.
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Question 503532: You have a wire that is 53 cm long. You wish to cut it into two pieces. One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle. Let A represent the total area of the square and the circle. What is the circumference of the circle when A is a minium?
You can put this solution on YOUR website! You have a wire that is 53 cm long. You wish to cut it into two pieces.
One piece will be bent into the shape of a square. The other piece will be bent into the shape of a circle.
Let A represent the total area of the square and the circle.
What is the circumference of the circle when A is a minimum?
:
Let x = circumference of the circle
then
(53-x) = the perimeter of the square
and = the side of the square
and = the area of the square
:
Find the area of the circle using the circumference
find the radius (r)
2*pi*r = x
r =
r =
Find the area of the circle
A =
Replace r with
A = * = *
cancel pi into 39.48
A =
Total area of circle and square
A = + = +
:
convert these fractions to decimal coefficients
A(x) = .0796x^2 + .0625x^2 - 6.625x + 175.5625
A(x) = .1421x^2 - 6.625x + 175.5625
:
Find the axis of symmetry of this quadratic equation (min area)
x =
x =
x = 23.31 cm is the circumference when they have min area
