You are designing an athletic field in the shape of a rectangle x meters long capped at two ends by semicircular regions of radius r. The boundary of the field is to be a 400 meter track. What values of x and r will give the rectangle its greatest area?

x  = 200 m

r  =  100 m

Step-by-step explanation:

Let call "x" one of the sides of the rectangle (the one finishing in semicircular areas then as "r" is the radius of the semicircular areas

x  + 2*r = 400   ⇒  2*r  =( 400 - x )

Area of the rectangle is:

A(r) = x*y           y  =  2*r

Then the area of the rectangle as a function of x is:

A(x) = x* ( 400 - x )    ⇒   A(x) = 400*x  -  x²

Taking derivatives on both sides of the equation we get:

A´(x)  =  400 -2*x

A´(x)  =  0       ⇒    400 -2*x  = 0

2*x  =  400

x =  200 m

And  r is equal to:

r  =  ( 400 - x ) / 2

r =  ( 400  - 200 ) / 2

r =  100 m  

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step-by-step explanation:

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