Question 737399
A lifeguard is interested in enclosing a rectangular swimming area along the beach.
 The swimming area will be enclosed using 72m of floating rope and will be bordered on one side by the beach: that is the problem these are the questions
:
a. Draw a sketch to model the situation
I'll let you handle this
:
b. Find the dimensions (length and width) that provide the maximum area for swimming.
Only 3 sides are required therefore
L + 2W = 72
L = (72-2W)
:
c. What is the maximum area available for swimming?
A = L * W
Replace L with (72-2W)
A = W(72-2W)
A quadratic equation
A = -2W^2 + 72W
Max area occurs on the axis of symmetry, find it using formula: x = -b/(2a)
W = {{{(-72)/(2*-2)}}}
W = 18 m is the width that gives max area
then
L = 72 - 2(18) = 36 m is the length for max area
:
Actual max area: 18 * 36 = 648 sq/m
:
:
If you graph the equation you can see that max area occurs when x=18 (width)
{{{ graph( 300, 200, -10, 50, -100, 700, -2x^2+72x) }}}