What Determines the Maximum Area of an Athletic Field?

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Homework Help Overview

The problem involves determining the maximum area of an athletic field designed as a rectangle with semicircular ends, constrained by a 400-meter racetrack. Participants are tasked with expressing the area in terms of either the rectangle's length or the semicircles' radius and identifying the dimensions that maximize this area.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • One participant attempts to express the area in terms of the radius, arriving at an equation but faces questions regarding the correctness of their approach and the definition of "overall area." Others seek clarification on how the initial area was calculated and challenge the validity of the derived expression.

Discussion Status

The discussion is ongoing, with participants questioning the assumptions made in the calculations and seeking further clarification on the steps taken to arrive at the area expression. There is no consensus yet, as participants are exploring different interpretations of the problem and the calculations involved.

Contextual Notes

Participants are working under the constraints of the problem statement, specifically the requirement to express the area based on the given dimensions and the total boundary defined by the racetrack. There is uncertainty regarding the initial calculations and definitions used in the area expression.

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Homework Statement


An athletic field is to be built in the sahpe of a rectangle x units long capped by semicircular regions of radius r at the two ends. The field is to be bounded by a 400-m racetrack.
a. Express the area of the rectangular portion of the field as a funcion of x alone or r alone (your choice).
b. What values of x and r give the rectangular portion the largest possible area?


The Attempt at a Solution


For a, i expressed the equation in terms of r. I got 40000/pi - pi(r)^2. i just took the overall area and subtract it by the semicircular circles.
 
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What is your question? If you are asking if you did this correctly, I can't say because didn't show how you got that answer. Are you saying that the "overall area" is 4000/pi? How did you get that?
 
i got the 40000/pi - pi(r)^2 by getting the overall area and minus it by the semicircular circles that cap the rectangular part. that's the answer for a.
 
dmonlama said:
i got the 40000/pi - pi(r)^2 by getting the overall area and minus it by the semicircular circles that cap the rectangular part. that's the answer for a.

No, it's not. If r=0 that gives 40000/pi for the area, which can't be right. Again, show how you reached that conclusion.
 
HallsofIvy said:
What is your question? If you are asking if you did this correctly, I can't say because didn't show how you got that answer. Are you saying that the "overall area" is 4000/pi? How did you get that?

dmonlama said:
i got the 40000/pi - pi(r)^2 by getting the overall area and minus it by the semicircular circles that cap the rectangular part. that's the answer for a.
Now, please answer my original question. HOW did you get "the overall area"?
 

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