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.)
Accepted Solution
A:
Answer:Dimensions :x = 625 ydy = 1250 ydA (max) = 781250 yd²Step-by-step explanation:Let "x" be the small side of the rectangle, and "y" the longerA = x*y perimeter is 2500 yd = 2x + y Then y = 2500 - 2x (1)A(x) = x * ( 2500 - 2x ) ⇒ A(x) = 2500x - 2x²Taking derivatives :A´(x) = 2500 - 4x and A´(x) = 02500 - 4x = 0 4x = 2500 x = 2500/4x = 625 ydNow by substtution of x value in equatio (1)y = 2500 - 2x ⇒ y = 2500 - 2* 625y = 2500 - 1250y = 1250 ydAnd fnally th aea is:A (max) = 1250 * 625 A (max) = 781250 yd²