Answer to Question #265842 in Calculus for Nicey

Question

A resort owner wants to enclose a beachfront area for swimming activities. Based on her plan, only 3 sides will be fence with 270 meter rope and floats, while the shoreline part will be open. Determine the dimension of the 3 sides of the rectangle that will give a maximum area.

Answer 1

Lets draw a picture

We have to find such value 0 < x < 270 that maximize the area of the blue rectangle. Let S be the area, then $S(x)=x*(270-2x)=270x-2x^2$

Lets find the critical points

$S'(x)=270-4x$

$270-4x=0\implies x =67.5$. Since to the left from x = 67.5 the value of S'(X) is positive, and to the right it's negative, then x = 67.5 is the point of maximum

The sides of a rectangle is 67.5 and 270 - 2 * 67.5 = 135 meters and the area is$S(67.5)=270*67.5-2*67.5^2=9112.5$ square meters