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 December 7th, 2017, 05:15 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, 06:09 PM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,641 Thanks: 1475 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, 07:16 PM #3 Senior Member   Joined: Sep 2015 From: USA Posts: 2,641 Thanks: 1475 oops.. all $w's$ should be $W's$ Thanks from skeeter December 7th, 2017, 07:47 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, 10:23 PM   #5
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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! Tags optimization, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shouav1 Calculus 1 November 28th, 2012 06:10 PM eveveve Calculus 1 August 5th, 2012 12:57 PM jkim28 Calculus 3 October 31st, 2009 08:20 AM a.a Calculus 1 August 26th, 2008 09:32 AM jokerthief Algebra 2 December 21st, 2007 01:26 AM

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