describe the largest pen area possible using the same amount of fencing in 6'X14' area. How do the perimeter and area compare to the pen above?
the largest area for a given perimeter is a square.
Since the perimeter of a 6x14 pen is 40, the side of a square would be 10.
10x10 = 100
6x14 = 84
thank you
To find the largest pen area possible using the same amount of fencing in a 6'x14' area, we will consider different shapes and dimensions.
1. Rectangle: Let's start with a rectangle shape. The perimeter of a rectangle is given by the formula P = 2 * (length + width). For a 6'x14' rectangle, the perimeter would be P = 2 * (6 + 14) = 40'. To maximize the area, we want a square shape. Since the perimeter is fixed at 40', we can distribute it equally among the four sides. So, each side of the square will have a length of 10' (40' / 4). Therefore, the largest pen area possible with a rectangle is 10'x10' = 100 square feet.
2. Circle: Let's consider a circular shape for the pen. To find the circumference of a circle, we use the formula C = 2 * π * r, where r is the radius of the circle. Since the perimeter is fixed at 40', we have 40' = 2 * π * r. Solving for r, we find r ≈ 6.37'. The largest pen area possible with a circle is given by the formula A = π * r^2, where A represents the area. Plugging in the value of r, we get A ≈ π * (6.37')^2 ≈ 127 square feet.
Comparing the results:
- The largest pen area possible with a rectangle is 100 square feet.
- The largest pen area possible with a circle is approximately 127 square feet.
Therefore, the pen area with the circular shape is greater than the rectangular shape when using the same amount of fencing.