A piece of wire 12 m long is cut into 2 pieces. One piece is bent into the shape of an equilateral triangle and the other into a shape of a circle. How should the wire be cut so that the combined area of the two figures is as small as possible?
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To find the optimal way to cut the wire so that the combined area of the two figures is as small as possible, we need to consider the formulas for the area of an equilateral triangle and a circle.
First, let's denote the length of the wire used for the equilateral triangle as "x" and the length of the wire used for the circle as "12 - x" (since the total length of the wire is 12 m).
The formula for the area of an equilateral triangle is given by:
Area of an equilateral triangle = (sqrt(3)/4) * side length^2
And the formula for the area of a circle is given by:
Area of a circle = π * radius^2
To minimize the combined area, we need to minimize the sum of the areas of the equilateral triangle and the circle. Let's call this combined area A:
A = (sqrt(3)/4) * x^2 + π * ((12 - x)/(2π))^2
Simplifying the equation:
A = (sqrt(3)/4) * x^2 + (12 - x)^2 / (4π)
Now, we can take the derivative of A with respect to x and solve for x to find the value of x that minimizes A:
dA/dx = (sqrt(3)/2) * x - (12 - x) / (2π)
Setting the derivative equal to zero:
(sqrt(3)/2) * x - (12 - x) / (2π) = 0
Simplifying:
sqrt(3) * x - (12 - x) / π = 0
Multiplying through by π:
sqrt(3) * π * x - 12 + x = 0
(x + sqrt(3) * π * x) = 12
Factoring out x:
x (1 + sqrt(3) * π) = 12
x = 12 / (1 + sqrt(3) * π)
Now that we have the value of x, we can calculate the corresponding value of (12 - x) to determine how the wire should be cut.
Once the wire is cut into these lengths, we can use the formulas for area to calculate the individual areas of the equilateral triangle and the circle.