Question

newton has 225 feet of fencing and wishes to enclose his englush bulldogs with a rectangular pen. what are the dimensions of the pen with the maximum area

Answer 1

To maximize the area of a rectangular pen with 225 feet of fencing, we should create a square pen. By dividing the total fencing by the number of sides in a square, we find that each side should be 56.25 feet, leading to a maximum area.

Step-by-step explanation:

The student's question is a classical optimization problem in Mathematics that involves finding the dimensions of a rectangle with the largest possible area given a fixed perimeter. This is a common problem in High School level geometry and algebra.

To achieve the maximum area with a fixed perimeter, we should make the pen a square since a square has the largest area for a given perimeter out of all rectangles. To find the dimensions of this square, we divide the total length of the fencing, 225 feet, by the number of sides of a square, which is 4. This gives us 225 / 4 = 56.25 feet. So, the dimensions of the pen with the maximum area that Newton can build are 56.25 feet by 56.25 feet.

Answer 2

The optimal rectangle dimensions to enclose the maximum area with 225 feet of fencing is a square with sides of 56.25 feet each, giving an area of 3164.0625 square feet.

Step-by-step explanation:

Optimal Dimensions for Maximum Area

To determine the dimensions of a rectangle with the maximum area that can be enclosed with a fixed perimeter, we can use calculus or recognize that a square would provide the maximum area. Given that Newton has 225 feet of fencing, if we enclose the bulldogs in a square-shaped pen, each side would have to be equal, thus dividing the total length of the fencing by 4 (since a square has four equal sides).

225 feet ÷ 4 = 56.25 feet. Therefore, each side of the square would be 56.25 feet long. However, if the shape does not have to be a square and a rectangle is preferred for practical purposes, one could choose dimensions that are factors of 225 and still result in a rectangle with a decent area. For example, dimensions of 75 feet by 37.5 feet.

However, for the maximum area, the square is the optimal shape. The area of the square would be 56.25 feet × 56.25 feet, which equals 3164.0625 square feet, and this would be the maximum area that could be enclosed with 225 feet of fencing.