
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 13th, 2017, 05:40 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 163 Thanks: 2  Partial Derivative Application about optimization of dimension
The bottom of a rectangular box costs twice as much per unit area as the sides and top. Find the shape for a given volume that will minimize cost. Any tips would be grateful. 
March 13th, 2017, 06:37 PM  #2  
Senior Member Joined: Jan 2017 From: Toronto Posts: 163 Thanks: 2  Quote:
(2V/3)^(1/3)* (2V/3)^(1/3) * (9V/4)^(1/3)  
March 13th, 2017, 07:28 PM  #3 
Member Joined: Oct 2016 From: Melbourne Posts: 77 Thanks: 35  
March 13th, 2017, 08:59 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 1,694 Thanks: 860  Quote:
we want to minimize $C = 2 l w + l w +2(lh + wh)$ subject to the constraint $V = l w h$ to do this with the Lagrange Multiplier method we solve $\nabla (C  \lambda (lwhV)) = \bf{0}$ chugging through it we get the set of equations $3w + 2h  w h \lambda = 0$ $3l + 2h  l h \lambda = 0$ $2(l+w)  l w \lambda = 0$ $l w h = V$ solving this is probably a bit of a mess but you end up with $l = w =\left(\dfrac{2V}{3}\right)^{1/3}$ $h = \left(\dfrac {3V}{ 2} \right)^{2/3}$  
March 14th, 2017, 04:02 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,922 Thanks: 785 
I would have done this slightly differently, writing the Lagrange multiplier equation as . That gives the three equations , and . Now, since a specific value for is not necessary for the solution, eliminate [tex]\lambda[/b] buy dividing one equation by another. Dividing the first equation by the second, . Dividing the first equation by the third, . We also have the condition . That gives three equations to solve for l, w, and h. 
March 14th, 2017, 05:21 AM  #6 
Newbie Joined: Mar 2017 From: South Africa Posts: 1 Thanks: 0 Math Focus: CALCULUS AND LINEAR ALGEBRA 
You have to make use of the 1st derivative and second derivative to obtain your desired answers


Tags 
application, derivative, dimension, optimizating, optimization, partial 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Extreme / FirstDerivative Test (Application Problem)?  POISONXMITHIE  Linear Algebra  1  October 20th, 2015 05:40 AM 
Derivative Application  hyperbola  Calculus  3  May 3rd, 2015 04:41 AM 
Application of derivative  sachinrajsharma  Calculus  1  May 18th, 2013 08:36 PM 
Application of derivative  tangents to latus rectum  sachinrajsharma  Calculus  3  May 8th, 2013 10:41 AM 
Application in the derivative ..?  EXPLORE  Calculus  2  January 28th, 2010 03:03 AM 