An indoor fitness room consists of a rectangular region with asemi-circle on eac

Question

An indoor fitness room consists of a rectangular region with asemi-circle on each end. The perimeter of the room is to be a200-meter running track. Let "x" be the length and "y" thewidth of the rectangular portion. 1. Determine the radius of the semicircle (should be1/2y). 2. Determine the distance in terms of "y" around the twosemicircular parts. 3. Use the result from 2 to write an equation interms of "x" and "y", for the distance traveled in one lap aroundthe track. 4. Use the result from 3 to write the area of therectangular region as a function of "x". 5. Graph the area function. What dimensions willproduce a maximum area of the rectangular region? An indoor fitness room consists of a rectangular region with asemi-circle on each end. The perimeter of the room is to be a200-meter running track. Let "x" be the length and "y" thewidth of the rectangular portion. 1. Determine the radius of the semicircle (should be1/2y). 2. Determine the distance in terms of "y" around the twosemicircular parts. 3. Use the result from 2 to write an equation interms of "x" and "y", for the distance traveled in one lap aroundthe track. 4. Use the result from 3 to write the area of therectangular region as a function of "x". 5. Graph the area function. What dimensions willproduce a maximum area of the rectangular region?

Explanation / Answer

Ok so the full perimeter of the entire shape is 200 meters causethat is the length of the track. The full perimeter equals 2x+2r because it is the 2 lengthsof the rectangle plus the two half-circle (or one whole circle) We know that the radius r=1/2y so we can say 2. The distance in terms of "y" around the semicircles should be p= 2 r = 2 (.5 y) = y 3. 200 = 2x + 2(.5y) = 2x + y 4. Area = 2x + 2y but from equ. 3 we know y = (200 - 2x)/ sothe Area = 2x + 2[(200 - 2x)/]

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