Question 1046437
A indoor track consists of a rectangular region with 2 semi-circles on the ends.
 The perimeter of the track is 200 meters.
 Write a function for the area of the track in terms of x and y, where x is the length of the rectangle and y is the height of the rectangle.
 Use the perimeter to write the area as just a function of y.
 Use your calculator to find the maximum area.
:

We know the diameter of the semi circles = the width of the rectangle, y
Therefore the circumference of the semi circles = pi*y
The total perimeter
{{{pi*y + 2x = 200}}}
{{{2x = 200 - pi*y}}}
divide both sides by 2
{{{x = 100 - pi*.5y}}}
:
total area = 2 semicircle area + rectangular potion area
A = {{{pi*(.5y)^2 + xy}}}
Replace x
A = {{{pi*(.5y)^2 + y(100-pi*.5y)}}} is the area as a function of y
:
Max area occurs when y = 63.66 m, then x=0, The track is a circle 
A = 3183 sq m