I need to get the largest area of 100 feet of rope. I want to say rectangle: 40L x 40L x 10w x 10w =100
Please advise if I am on the right track. Keeping in mind that you can use any Plan Geometry Shape?
A = length * width
40 * 10 = 400 sq. ft
Try a square and a circle.
http://www.mathsisfun.com/area.html
*--Considering all rectangles with the same perimeter, the square encloses the greatest area.
Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding a to one side and subtracting a from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.
*--Considering all rectangles with the same area, the square results in the smallest perimeter for a given area.
*--Considering all shapes, the circle encloses the maximum area for a given perimeter.
The geometric figure that encloses the most area is a circle.
A circumference C=100=2pi r so r=15.92 and
Area = pi r^2 = 795.77
Your rectangle of 40*10 = only 400 sq. ft.
A perimeter of 25 feet per side times 4 sides = a length of rope 100 feet long and encloses the greatest area for an oblong figure, of 25^2 = 625 feet.
Clearly, the circle is a vast improvement, by 795.77/625=127.32%
An ellipse is not more efficient than a circle.
The reason is that a circle uses only one leg or end to multiply by, whereas other figures involve multiple generators or legs or multipliers. A radius is fixed at one end. But when you multiply down a rectangle, you are really moving the width (with its TWO ENDS) along the entire Length (with its TWO ENDS)! That is less efficient. The saved energy shows up in the area conserved.
Yes, you are on the right track. To find the largest area using 100 feet of rope, you can indeed use any plan geometry shape. However, the shape you mentioned, a rectangle with sides of 40L, 40L, 10w, and 10w, does not seem to be correct.
To maximize the area, you need to consider shapes that have an equal distribution of the rope around all sides. This means that all sides of the shape should be as close to each other in length as possible.
The shape that satisfies these conditions is a square. A square has all sides equal in length, so you can divide the 100 feet of rope into four equal parts, each measuring 25 feet. By using these four equal lengths as the sides of the square, you can maximize the area enclosed by the shape.
To calculate the area of a square, you need to multiply the length of one side by itself (squared): Area = side length x side length.
In this case, the side length is 25 feet, so the area of the square would be 25 feet x 25 feet = 625 square feet. Therefore, a square with sides of 25 feet each will give you the maximum area using 100 feet of rope.
So, to answer your question, a square shape with sides measuring 25 feet each will give you the largest area of 100 feet of rope.