calculate the largest area possible that can be made by a 15 meter long wire that will be bent to form a closed plane area. No figure given
the largest area of course is contained within a circle.
So if the perimeter is 15 m
then 2πr = 15
r = 15/(2π)
area = π(15/(2π))^2
= 225/(4π) m^2 or appr 17.9 m^2
To find the largest possible area that can be made with a given length of wire, we need to determine the shape that will maximize the area.
Since the wire is bent to form a closed plane area, we can consider different shapes like a square, rectangle, circle, or even irregular polygons.
Let's analyze each case and see which one produces the largest area.
1. Square:
A square has four equal sides, so if the perimeter (wire length) is 15 meters, each side will have a length of 15/4 = 3.75 meters. The area of a square is calculated by multiplying the length of one side by itself, so the area of the square will be 3.75 * 3.75 = 14.06 square meters.
2. Rectangle:
Let's assume one side of the rectangle is longer than the other side. If the perimeter is still 15 meters, let's say the longer side is 7.5 meters and the shorter side is 15 - 2 * 7.5 = 0 meters (because it would be a line). The area of this rectangle would be 7.5 * 0 = 0 square meters.
This means that having one side with zero length would not create an enclosed plane area.
3. Circle:
A circle maximizes the area for a given perimeter. The equation that relates the circumference (perimeter) to the radius (half the diameter) of a circle is C = 2 * π * r, where C = 15 meters. Rearranging the equation to solve for r, we have r = C / (2 * π) = 15 / (2 * π) ≈ 2.39 meters. The area of the circle is calculated using the formula A = π * r^2, so the area would be π * (2.39)^2 ≈ 17.97 square meters.
4. Irregular Polygon:
For an irregular polygon, determining the maximum area would involve complex mathematical calculations, which may not be feasible without additional specifications or constraints.
Based on the analysis above, it is clear that the circle yields the largest possible area, with an approximate area of 17.97 square meters.