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

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> 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       Log On


   

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) About Me  (Show Source):
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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%2Ay+%2B+2x+=+200
2x+=+200+-+pi%2Ay
divide both sides by 2
x+=+100+-+pi%2A.5y
:
total area = 2 semicircle area + rectangular potion area
A = pi%2A%28.5y%29%5E2+%2B+xy
Replace x
A = pi%2A%28.5y%29%5E2+%2B+y%28100-pi%2A.5y%29 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