SOLUTION: 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

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.
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
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

divide both sides by 2

:
total area = 2 semicircle area + rectangular potion area
A =
Replace x
A = 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

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