A piece of wire of 14 meters long is cute into 2 pieces. One piece is bent into the shape of an equilateral triangle and the other into the shape of a circle. How should be the wire be cut so that the combined area of 2 figures is as small as possible?
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To solve this problem, we need to find out how to divide the wire into two pieces so that the combined area of the equilateral triangle and the circle is as small as possible.
Let's assume that the length of the wire used for the equilateral triangle is x. Therefore, the length of the wire used for the circle is (14 - x).
The formula for the perimeter of an equilateral triangle is P = 3s, where s is the length of each side. Since the equilateral triangle is formed from x units of wire, we can derive the following equation:
x = 3s
To find the area of an equilateral triangle, we can use the formula A = (sqrt(3)/4) * s^2.
So, the area of the equilateral triangle is A1 = (sqrt(3)/4) * x^2.
Moving on to the circle, the formula for the circumference of a circle is C = 2πr, where r is the radius. Since the circle is formed from (14 - x) units of wire, we can derive the following equation:
(14 - x) = 2πr
To find the area of a circle, we can use the formula A = πr^2. Thus, the area of the circle is A2 = πr^2.
Now, we need to eliminate either s or r from the equations to create an equation with a single variable. We can do this by substituting s in terms of x or r in terms of x. Since the equation for the equilateral triangle already has x in it, let's solve for r in terms of x from the equation for the circle.
(14 - x) = 2πr
r = (14 - x) / (2π)
Now, substitute this value of r into the formula for the area of the circle to get a single variable equation:
A2 = π((14 - x) / (2π))^2
A2 = (14 - x)^2 / (4π)
We can now calculate the total area, A_total, of the equilateral triangle and the circle by adding A1 and A2:
A_total = A1 + A2
A_total = (sqrt(3)/4) * x^2 + (14 - x)^2 / (4π)
To find the value of x that minimizes the combined area, we can take the derivative of A_total with respect to x and set it to zero:
d(A_total) / dx = 0
Differentiating the equation gives us:
(√3/2) * x - (7 - x) / (4π) = 0
Simplifying further, we get:
(√3/2) * x - (7/4π) + (x/4π) = 0
Rearranging the equation, we find:
(5/4π) * x = (7/4π)
Simplifying, we get:
x = (7/5) * π
Since x represents the length of the wire used for the equilateral triangle, the length of the wire used for the circle is:
14 - x = 14 - (7/5) * π
Therefore, the wire should be cut into approximately (7/5) * π units for the equilateral triangle and 14 - (7/5) * π units for the circle in order to minimize the combined area of the two shapes.