A wire 9 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each:
For the square?
For the circle?
Where should the wire be cut to maximize the total area? Again, give the length of wire used for each:
For the square?
For the circle?
To minimize the total area, the wire should be cut into two equal pieces.
For the square:
The perimeter of a square is given by 4 * side length. Since the wire is 9 meters long, each side of the square will have a length of 9 / 4 = 2.25 meters. Therefore, the wire used for the square will be 4 * 2.25 = 9 meters.
For the circle:
The circumference of a circle is given by 2 * π * radius. The wire used for the square is 9 meters, so the radius can be found by dividing it by 2 * π = 6.28. Therefore, the radius of the circle will be 9 / 6.28 ≈ 1.43 meters. The wire used for the circle will be 2 * π * 1.43 ≈ 8.98 meters.
To maximize the total area, the wire should be cut into two unequal pieces.
For the square:
To maximize the area of the square, the wire should be cut into three smaller pieces: 1 meter, 1 meter, and 7 meters. The first two pieces will be used to form two sides of the square, each with a length of 1 meter, and the remaining 7 meters will be used for the other two sides. Therefore, the wire used for the square will be 1 + 1 + 7 = 9 meters.
For the circle:
The remaining 7 meters of the wire will be used to form the circumference of the circle. The radius can be found by dividing the remaining wire by 2 * π = 6.28. Therefore, the radius of the circle will be 7 / 6.28 ≈ 1.11 meters. The wire used for the circle will be 2 * π * 1.11 ≈ 6.97 meters.
To minimize the total area, we want to cut the wire in a way that maximizes the area of the square and minimizes the area of the circle.
Let x represent the length of wire used for the square. Since a square has four equal sides, each side of the square will have a length of (x/4). The perimeter of the square is equal to the total length of wire used, so we have:
4 * (x/4) = 9
x = 9/4
x = 2.25 meters
Therefore, we should cut the wire into a length of 2.25 meters for the square.
Since the remaining length of wire after cutting x for the square is (9 - 2.25) = 6.75 meters, we will use this length for the circle.
For the circle, the circumference is equal to the total length of wire used. We can use the formula for the circumference of a circle:
2 * r * π = 6.75
Solving for r:
r = (6.75 / (2 * π))
We can then find the area of the circle using the formula:
A = π * r^2
Substituting the value of r, we get:
A = π * ((6.75 / (2 * π))^2) = 3.393 m^2 (rounded to three decimal places)
Therefore, we should cut the wire into a length of 6.75 meters for the circle.
To maximize the total area, we want to cut the wire in a way that maximizes the area of both the square and the circle.
Since the perimeter of the square is fixed at 9 meters, we will cut the wire in half, resulting in x = (9 / 2) = 4.5 meters for the square.
The remaining wire length is also 4.5 meters, which we will use for the circle.
For the circle, we use the same approach as before:
2 * r * π = 4.5
Solving for r:
r = (4.5 / (2 * π))
Finding the area of the circle:
A = π * ((4.5 / (2 * π))^2) = 1.273 m^2 (rounded to three decimal places)
Therefore, we should cut the wire into a length of 4.5 meters for the square and also 4.5 meters for the circle to maximize the total area.