Question

A lifeguard marks off a rectangular swimming area at a beach480 m of rope. What is the greatest possible area she can enclose?

Answer 1

Either `28,800` or `14,400` square meters depending upon the interpretation of the description.

Explanation:

Possibility 1: The beach forms one side of the rectangular area (no rope required)

If `L` represents the length of the side paralleling the beach
and `W` represents the with of the remaining two sides perpendicular to the beach
then
`color(white)("XXX")L=480-2Wcolor(white)("xxxxx")`(all measurements in meters)
and the area would be
`color(white)("XXX")A_(L,W)= LxxW`
or
`color(white)("XXX")A(W)=480W-2W^2`

The maximum value for `A(W)` would be achieved when the derivative `A'(W)=0`

`color(white)("XXX")A'(W)=480-4W=0`

`color(white)("XXX")rArr W=120`

and, since `L=480-2W`
`color(white)("XXX")rArr L=240`

Giving a total possible area of
`color(white)("XXX")LxxW= 240xx120=28,800` (square meters)

Possibility 2: All 4 sides require rope
In this case the maximum area is formed by a square with sides of length `480/4=120` (meters)
and
a (maximum) area of `120xx120=14400` square meters