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December 7th, 2017, 05:15 PM   #1
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Optimization Problem

Basically all I am struggling with is finding an equation for the function 𝒇(𝒙) giving the quantity that is to be maximized as a function of a variable. The rest of the problem is more complex but should be easy once I figure the first part out.

The Problem:
A park is to be in the shape of a rectangle, with a perimeter of 760 feet. The park is to be divided into 3 rectangular regions of equal size by two parallel walkways of equal width. Each of the 3 rectangular regions is to have an area of 9000 square feet. Design the park so that each walkway is as wide as possible.
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December 7th, 2017, 06:09 PM   #2
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Let $W$ be the width of the park and $\delta$ the width of a pathway.

The park will have perimeter

$P = 760 = \dfrac{54000}{W} + 4 \delta + 2W$

$\delta = \dfrac{760 - 2W-\frac{54000}{W}}{4}$

$\dfrac{d\delta}{dW} = -\dfrac 1 2 +\dfrac{13500}{w^2}$

$\dfrac{d\delta}{dW} = 0 \Rightarrow \dfrac{13500}{w^2}= \dfrac 1 2$

$w^2 = 27000$

$w = 30\sqrt{30}$

$\delta = 190-30 \sqrt{30} \approx 25.68$
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December 7th, 2017, 07:16 PM   #3
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oops.. all $w's$ should be $W's$
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December 7th, 2017, 07:47 PM   #4
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Quote:
Originally Posted by romsek View Post
oops.. all $w's$ should be $W's$
I hate when that happens ...
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December 8th, 2017, 10:23 PM   #5
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Quote:
Originally Posted by romsek View Post
Let $W$ be the width of the park and $\delta$ the width of a pathway.

The park will have perimeter

$P = 760 = \dfrac{54000}{W} + 4 \delta + 2W$

$\delta = \dfrac{760 - 2W-\frac{54000}{W}}{4}$

$\dfrac{d\delta}{dW} = -\dfrac 1 2 +\dfrac{13500}{w^2}$

$\dfrac{d\delta}{dW} = 0 \Rightarrow \dfrac{13500}{w^2}= \dfrac 1 2$

$w^2 = 27000$

$w = 30\sqrt{30}$

$\delta = 190-30 \sqrt{30} \approx 25.68$
thank you so much! this helped tremendously!
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