You have a wire that is 89 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?
To find the circumference of the circle when A is a minimum, we need to apply calculus concepts.
Let's denote the side length of the square as x. Since all sides of a square are equal, the perimeter of the square is 4x.
The remaining wire length, after cutting out the square, will be 89 - 4x. This remaining length will be used to form the circumference of the circle.
Let's denote the radius of the circle as r. The formula for the circumference of a circle is 2πr, where π is a constant (approximately 3.14159).
The remaining wire length, 89 - 4x, is equal to the circumference of the circle: 2πr = 89 - 4x.
Now, let's express the area A in terms of x and r.
The area of the square is calculated as A_square = (side length)^2 = x^2.
The area of the circle is calculated as A_circle = πr^2.
The total area A is the sum of the areas of the square and the circle: A = x^2 + πr^2.
To find the value of x and r that minimizes A, we need to take the derivative of A with respect to both x and r, set the derivatives equal to zero, and solve for x and r.
dA/dx = 2x
dA/dr = 2πr
Now, setting dA/dx and dA/dr equal to zero:
2x = 0
2πr = 0
Solving for x and r:
x = 0
r = 0
However, x = 0 is not applicable since it will result in a degenerate square with zero area. Therefore, we need to find the value of x and r that satisfy the equation 2πr = 89 - 4x.
Substituting r = (89 - 4x) / (2π) into the equation A = x^2 + πr^2:
A = x^2 + π[(89 - 4x) / (2π)]^2
Simplifying:
A = x^2 + [(89 - 4x)^2] / (4π)
Now, taking the derivative of A with respect to x and setting it equal to zero to find the minimum:
dA/dx = 2x - 2[(89 - 4x)(4)] / (4π) = 0
Simplifying:
2x - 8(89 - 4x) / (4π) = 0
2x - 8(89 - 4x) = 0
2x - 712 + 32x = 0
34x = 712
x = 712 / 34
x ≈ 20.941
Now, substituting this value of x into the equation 2πr = 89 - 4x to find r:
2πr = 89 - 4(20.941)
2πr = 89 - 83.764
2πr ≈ 5.236
r ≈ 5.236 / (2π)
r ≈ 0.833
Finally, to find the circumference of the circle when A is at a minimum, we use the formula for the circumference:
C = 2πr ≈ 2π(0.833) ≈ 5.233 cm (rounded to three decimal places).
Therefore, when A is a minimum, the circumference of the circle is approximately 5.233 cm.
To determine the circumference of the circle when the total area (A) is minimum, we need to find the relationship between the length of the wire and the areas of the square and the circle.
Let's assume that one piece of the wire is x cm long, and the other piece is (89 - x) cm long. We will use x to represent the length of the wire used for the square, and (89 - x) as the length of the wire used for the circle.
Since a square has four equal sides, the perimeter of the square (P) is equal to 4 times the length of one side, which is x. Therefore, P = 4x.
To find the side length (s) of the square, we use the formula: s = x/4, since the perimeter of a square is the sum of all sides.
The area of the square (A_s) is given by the formula A_s = s^2.
Now, for the circle, the circumference (C) is given by the formula: C = 2πr, where r is the radius of the circle.
To relate the circumference and the length of the wire used for the circle, we have 2πr = (89 - x).
The area of the circle (A_c) is given by the formula A_c = πr^2.
Since A represents the total area of the square and the circle, we have A = A_s + A_c.
Substituting the values we found for A_s and A_c, we get A = (x/4)^2 + πr^2.
To find the minimum value of A, we can take the derivative of A with respect to x, set it equal to zero, and solve for x.
dA/dx = 2x/16 + 2πr(dr/dx) = 0.
Simplifying the equation, we get x/8 + πr(dr/dx) = 0.
Since we are looking for the minimum value of A, we need to find the value of x that satisfies this equation.
Now, we need to use the relationship between x and r to solve the equation. We know that 2πr = 89 - x.
Substituting this relationship into the equation, we get x/8 + (89 - x)(dr/dx) = 0.
Simplifying further, (dr/dx) = -x/(8(89 - x)).
Now, we can substitute the value of (dr/dx) into the previous equation:
-x/(8(89 - x)) = 0.
Solving the equation, we find that x = 0 or x = 89.
Since the wire cannot be of length 0, we discard that solution.
Therefore, the only feasible solution is x = 89.
Substituting this value into the equation for the circumference of the circle, we have:
C = 2πr = 2π(89 - x) = 2π(89 - 89) = 0.
Thus, when A is a minimum, the circumference of the circle is 0.
Note: Please double-check the calculations as they are subject to human error.
Let the circle have radius r, and the square have side s. Then
2πr + 4s = 89
A = πr^2 + s^2 = πr^2 + (89-2πr)^2
A = (4π^2+π)r^2 - 356πr + 7921
A is minimum when r = 178/(4π+1)
C = 2πr = 356π/(4π+1)