You have a wire that is 71 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?
The circumference of the circle is ? cm
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To find the circumference of the circle when the total area (A) is at a minimum, we need to set up an equation and differentiate it with respect to the circumference to find the critical point where the area is minimum.
Let's start by assigning variables to the unknowns:
x = side length of the square (in cm)
r = radius of the circle (in cm)
Given:
Total length of the wire = 71 cm
Since the wire is used to form both a square and a circle, we can derive equations for the perimeter of the square and the circumference of the circle:
Perimeter of the square = 4x
Circumference of the circle = 2πr
Using the given information, we can create an equation to express the total length of the wire:
4x + 2πr = 71
To find the minimum area, we need to express the total area (A) in terms of x.
Area of the square = x^2
Area of the circle = πr^2
Total area (A) = x^2 + πr^2
Now, we need to eliminate one variable to express A in terms of x alone. Rearrange the perimeter equation to express r in terms of x:
r = (71 - 4x) / (2π)
Substitute this value of r into the equation for A:
A = x^2 + π [(71 - 4x) / (2π)]^2
Simplifying:
A = x^2 + [(71 - 4x) / 2]^2
To find the minimum area, we differentiate A with respect to x and set the derivative equal to zero:
dA/dx = 2x - 2(71 - 4x) / 2
dA/dx = 2x - (71 - 4x)
dA/dx = 6x - 71
Setting dA/dx equal to zero:
6x - 71 = 0
6x = 71
x = 71 / 6
Substituting this value of x back into the perimeter equation:
r = (71 - 4 * (71 / 6)) / (2π)
r = 71 / (12π)
We now have the values of x and r. To find the circumference of the circle at minimum area (A), substitute the value of r into the circumference equation:
Circumference of the circle = 2π * (71 / (12π))
Circumference of the circle = 71 / 6
Therefore, the circumference of the circle when A is at a minimum is 71/6 cm.