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

Mar 4, 2018

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
$\textcolor{w h i t e}{\text{XXX")L=480-2Wcolor(white)("xxxxx}}$(all measurements in meters)
and the area would be
$\textcolor{w h i t e}{\text{XXX}} {A}_{L , W} = L \times W$
or
$\textcolor{w h i t e}{\text{XXX}} A \left(W\right) = 480 W - 2 {W}^{2}$

The maximum value for $A \left(W\right)$ would be achieved when the derivative $A ' \left(W\right) = 0$

$\textcolor{w h i t e}{\text{XXX}} A ' \left(W\right) = 480 - 4 W = 0$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow W = 120$

and, since $L = 480 - 2 W$
$\textcolor{w h i t e}{\text{XXX}} \Rightarrow L = 240$

Giving a total possible area of
$\textcolor{w h i t e}{\text{XXX}} L \times W = 240 \times 120 = 28 , 800$ (square meters)

Possibility 2: All 4 sides require rope
In this case the maximum area is formed by a square with sides of length $\frac{480}{4} = 120$ (meters)
and
a (maximum) area of $120 \times 120 = 14400$ square meters