My Math Forum Optimization Problem

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 December 7th, 2017, 04:15 PM #1 Newbie   Joined: Dec 2017 From: Ironton OH Posts: 2 Thanks: 0 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.
 December 7th, 2017, 05:09 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,500 Thanks: 1372 Let $W$ be the width of the park and $\delta$ the width of a pathway. The park will have perimeter $P = 760 = \dfrac{54000}{W} + 4 \delta + 2W$ $\delta = \dfrac{760 - 2W-\frac{54000}{W}}{4}$ $\dfrac{d\delta}{dW} = -\dfrac 1 2 +\dfrac{13500}{w^2}$ $\dfrac{d\delta}{dW} = 0 \Rightarrow \dfrac{13500}{w^2}= \dfrac 1 2$ $w^2 = 27000$ $w = 30\sqrt{30}$ $\delta = 190-30 \sqrt{30} \approx 25.68$ Thanks from skeeter and codyrawlins
 December 7th, 2017, 06:16 PM #3 Senior Member     Joined: Sep 2015 From: USA Posts: 2,500 Thanks: 1372 oops.. all $w's$ should be $W's$ Thanks from skeeter
December 7th, 2017, 06:47 PM   #4
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Quote:
 Originally Posted by romsek oops.. all $w's$ should be $W's$
I hate when that happens ...

December 8th, 2017, 09:23 PM   #5
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Joined: Dec 2017
From: Ironton OH

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Quote:
 Originally Posted by romsek 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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