Question

A renovation company donates 80 m of fence to a day-care center to build a rectangular play area for the children. Determine the dimension of the play area that would produce the largest possible area. (Assume that the fence can be cut to suit measurements.)

Answer 1

The largest area for the day-care center's play area is achieved by forming a square with the donated 80 m of fence. The dimensions should be 20 m by 20 m, giving an area of 400 m².

Step-by-step explanation:

To determine the dimensions of the play area that would produce the largest possible area using 80 m of fence we need to maximize the enclosed area of a rectangle with a given perimeter. This is a classic optimization problem in mathematics, which states that for a given perimeter, a rectangle has the largest area when it is a square. Therefore, to maximize the area, the daycare center should use the donated fencing to create a square with each side measuring 20 m (since the perimeter is 80 m, and 80 m divided by 4 sides of a square gives us 20 m per side).

The calculation is simple:
Perimeter of a square = 4 × side length
80 m = 4 × side length
side length = 80 m / 4
side length = 20 m

So, the play area will have the dimensions of 20 m by 20 m, providing an area of 400 m², which is the largest area that can be enclosed.