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November 2nd, 2016, 11:17 AM  #1 
Newbie Joined: Nov 2016 From: The world Posts: 2 Thanks: 0  Optimization
Hello! I am having problems solving the following exercise: An athletics track has two ‘straights’ of length l m and two semicircular ends of radius x m. The perimeter of the track is 400 m. a. Show that l = 200  π(pi)x and hence write down the possible values that x may have. b. What values of l and x maximise the shaded rectangle inside the track? What is this maximum area? ￼￼￼￼￼￼ Last edited by musiclover; November 2nd, 2016 at 11:21 AM. 
November 2nd, 2016, 11:42 AM  #2  
Senior Member Joined: Sep 2015 From: USA Posts: 1,646 Thanks: 836  Quote:
$\ell = \dfrac 1 2 (400  2 \pi x) = 200  \pi x$ $\ell \geq 0 \Rightarrow 0 \leq x \leq \dfrac{200}{\pi}$ You want to maximize $A = x \ell = x (200  \pi x) = 200x  \pi x^2$ we do this is usual way, solve $\dfrac{dA}{dx} = 0$ for $x$ $\dfrac{dA}{dx} = 200  2\pi x = 0$ $x = \dfrac{100}{\pi}$ $\ell = 200  \pi\left(\dfrac{100}{\pi}\right) = 100$ $A = \dfrac{10000}{\pi}$  

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