Maximizing Area of Rectangle w/440yd Perimeter

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SUMMARY

The discussion centers on maximizing the area of a rectangle with a fixed perimeter of 440 yards, incorporating semicircles at each end. The user initially set up the equations correctly, using the perimeter equation \(440 = 2x + 2\pi r\) and the area equation \(A = \pi r^2 + 2rx\). However, the user miscalculated the area by not correctly accounting for the dimensions of the rectangle and semicircles, leading to an incorrect radius of 70.03 yards instead of the correct value of 110 yards as per the textbook solution.

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  • Understanding of basic calculus, specifically derivatives
  • Familiarity with geometric properties of rectangles and semicircles
  • Knowledge of perimeter and area formulas for composite shapes
  • Ability to solve equations involving variables and constants
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  • Review calculus concepts related to optimization and derivatives
  • Study the properties of semicircles and their integration into rectangular dimensions
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Students studying calculus, geometry enthusiasts, and anyone involved in optimization problems related to area and perimeter in engineering or architecture.

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Homework Statement


A running track consists of a rectangle with a semicircle at each end. If the perimeter is to be exactly 440 yards, find the dimensions (x and r) that maximize the area of the rectangle. [Hint The perimeter is 2x + 2[tex]\pi[/tex]r



Homework Equations





The Attempt at a Solution


Ok I attempted this twice and got the exact same answer, twice. Here is what I did.

First I set up the equation: 440 = 2x + 2[tex]\pi[/tex]r

I then set up the equation: Area (total) = [tex]\pi[/tex]r2 + 2rx where x is the length of the side of the field (not counting the semicircles) and r is the radius.

I solved for x from the first equation and came up with x = 220 - [tex]\pi[/tex]r

I then plugged the value of x into the second equation. Once I destrubuted it, I took the derivative and set it to zero and had 2r[tex]\pi[/tex] + 440 - 4[tex]\pi[/tex]r = 0

Solving for r, I got 70.03. The answer in the back of the book is 110. What am I doing wrong?

I appreciate the help!
 
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Hi wcbryant87! :smile:
wcbryant87 said:
… maximize the area of the rectangle.

erm :redface: … wrong area! :wink:
 
haha. wow. the 'aha' moment has hit me. thanks!
 

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