
Differential Equations Ordinary and Partial Differential Equations Math Forum 
 LinkBack  Thread Tools  Display Modes 
November 2nd, 2016, 10:17 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 10:21 AM. 
November 2nd, 2016, 10:42 AM  #2  
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,490 Thanks: 749  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 $x$ $\dfrac{dA}{dx} = 200  2\pi x = 0$ $x = \dfrac{100}{\pi}$ $\ell = 200  \pi\left(\dfrac{100}{\pi}\right) = 100$ $A = \dfrac{10000}{\pi}$  

Tags 
differential, math, optimization 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
PSO OPtimization  markolive12  Applied Math  1  January 17th, 2014 07:47 AM 
Optimization  Renee  Algebra  0  November 30th, 2008 08:50 PM 
optimization help  pranavpuck  Calculus  1  November 30th, 2008 06:23 PM 
Optimization  squeeze101  Algebra  2  November 16th, 2008 03:51 PM 
Optimization help...  jeremic  Algebra  2  December 28th, 2006 07:06 PM 