If you have 100 feet of fence, which measurements would give you the largest area?
To find the measurements that would give you the largest area given a fixed amount of fence, you need to consider different scenarios and calculate the corresponding areas.
Let's assume you are given 100 feet of fence to enclose a rectangular area. Let's call the length of the rectangle L and the width W.
1. Start by considering extreme cases:
- If you make the length equal to 100 feet, the width would be 0 because all the fence length is used to create the length. However, an area with no width will be equal to 0.
- Similarly, if you make the width equal to 100 feet, the length would be 0, resulting in an area of 0.
2. Now, consider an equal division of the fence length:
- If you divide the fence length into two equal sides of 50 feet each, you would have a square with sides measuring 50 feet. The area of a square is calculated by multiplying the length of one side by itself. In this case, the area would be 50 feet x 50 feet = 2500 square feet.
3. Lastly, consider different length-to-width ratios:
- If you try different length-to-width ratios, you might find that the maximum area is achieved when the length and width are closer in value.
By comparing the different scenarios, you can determine that the largest area is obtained when the rectangular enclosure is in the form of a square, with sides measuring 50 feet each.
form the 100 ft into a circle.
If you have a rectangular field, a square gives the maximum area of 25^2 = 625.
A regular pentagon has an area of 688.
As you add more sides, a regular n-gon with more sides has greater area.
Reiny's circle of radius 50/π has the maximum possible area of 2500.