A lifeguard needs to rope off a rectangular swimming area in front of long lake beach, using 2500 yd of rope and floats. What dimensions of the rectangle will maximize the area? What is the maximum area? (Note that the shoreline is one side of the rectangle.)

Dimensions:

x = 625 yd

y = 1250 yd

A (max) = 781250 yd²

Step-by-step explanation:

Let "x" be the small side of the rectangle, and "y" the longer

A = x*y perimeter is 2500 yd = 2x + y

Then y = 2500 - 2x (1)

A (x) = x * (2500 - 2x) ⇒ A (x) = 2500x - 2x²

Taking derivatives:

A' (x) = 2500 - 4x and A' (x) = 0

2500 - 4x = 0 4x = 2500 x = 2500/4

x = 625 yd

Now by substtution of x value in equatio (1)

y = 2500 - 2x ⇒ y = 2500 - 2 * 625

y = 2500 - 1250

y = 1250 yd

And fnally th aea is:

A (max) = 1250 * 625

A (max) = 781250 yd²