You have a wire that is 65 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
s = side of square
r = radius of circle
We know that
4s + 2πr = 65
Now, the combined area
A = s^2 + πr^2 = ((65-2πr)/4)^2 + πr^2 = (π^2 + 4π)r^2 + 65π/4 r + 4225/16
This is just a parabola, with its vertex at r = -b/2a = 65/(2π+8)
so the value of A = 4225/(4π+16)
To find the circumference of the circle when the total area (A) is a minimum, we first need to express A in terms of the circumference.
Let's start by cutting the wire into two pieces. Let x represent the length of one piece of wire that will be used to form the square. The other piece of wire, with a length of (65 - x), will be used to form the circle.
The perimeter of the square will be 4x, and since each side of the square is equal in length, the length of each side will be x/4. Therefore, the area of the square can be expressed as A(square) = (x/4)^2 = x^2/16.
The circumference of the circle can be expressed as C = 2πr = 2π((65 - x)/(2π)) = (65 - x).
To find the total area (A), we need to sum the areas of the square and the circle: A = A(square) + A(circle) = x^2/16 + πr^2.
In order for A to be a minimum, we need to find the value of x that minimizes the total area A.
To do this, we can differentiate A with respect to x and set it equal to zero:
dA/dx = (2x)/16 + (2π(65 - x))/1 = 0.
Simplifying the expression:
x/8 + π(65 - x) = 0.
Solving for x:
x/8 + 65π - xπ = 0.
x - 8x/8 + 65π - xπ = 0.
7x/8 = 65π.
x = (8 * 65π) / 7.
Substituting this value of x back into the expression for the circumference of the circle, we can find its value:
C = 65 - x.
C = 65 - (8 * 65π) / 7.
Simplifying the expression:
C = (455 - 8 * 65π) / 7.
Therefore, the circumference of the circle when the total area A is a minimum is given by (455 - 8 * 65π) / 7.
To find the circumference of the circle when the total area A is minimum, we first need to determine the relationship between the lengths of the wire used for the square and the circle.
Let's denote the length of the wire used for the square as x. Since the wire is cut into two pieces, the length of the wire used for the circle would be (65 - x).
For the square, all four sides have the same length, so each side of the square would be x/4. Therefore, the area of the square can be calculated as A_square = (x/4)^2 = x^2/16.
For the circle, the circumference (perimeter) can be calculated using the formula C_circle = 2πr, where r is the radius. Since the wire is used to form the entire circle, the circumference would be equal to the length of wire (65 - x).
To find the minimum value of A, we need to find the value of x that minimizes A, which means finding the value of x where the derivative of A with respect to x is equal to zero.
Taking the derivative of A with respect to x, we have:
dA/dx = 2x/16
Setting this derivative equal to zero, we get:
2x/16 = 0
x = 0
However, x cannot equal zero because we need to have a positive length of wire for both the square and the circle.
So, we need to find x where dA/dx is undefined. The only situation where the derivative is undefined is when x is equal to 0 or when x is equal to 65. However, x = 65 would result in an infinitely small circle, so it is not a valid solution.
Therefore, the only value for x that can make A a minimum is x = 0, which means using all the wire for the circle and none for the square. In this case, the circumference of the circle would be 65 cm.
So, when A is a minimum, the circumference of the circle is 65 cm.