User Name Remember Me? Password

 Calculus Calculus Math Forum

 July 27th, 2017, 07:22 AM #1 Senior Member   Joined: Nov 2015 From: United States of America Posts: 198 Thanks: 25 Math Focus: Calculus and Physics Minimize surface area for tank Hello forum. I am trying to make a rectangular tank that has a volume of 500 cubic feet, but minimizes surface area. The rectangular tank does not have a lid. $V = xyz = 500$ $\Rightarrow z = \frac{500}{xy}$ Surface area can be described as: $S = 2yz + 2xz + xy$ This accounts for the walls and base, but not the lid. Plugging in the constraint formula. $S = 2y(\frac{500}{xy}) + 2x(\frac{500}{xy}) + xy$ $\Rightarrow \frac{1000}{x} + \frac{1000}{y} + xy$ This is now a function with respect to two variables. $S(x,y) = \frac{1000}{x} + \frac{1000}{y} + xy$ $\frac{\partial S}{\partial x}$ = $-\frac{1000}{x^2} + y$ $\frac{\partial S}{\partial y}$ = $-\frac{1000}{y^2} + x$ Setting these partials equal to $0$. $-\frac{1000}{x^2} + y = 0$ $\Rightarrow y = \frac{1000}{x^2}$ and... $-\frac{1000}{y^2} + x = 0$ $\Rightarrow x = \frac{1000}{y^2}$ At this point I hit a wall. I struggle in actually finding the critical point here. It seems that x and y will have the same values though. Perhaps my tank should be a cube. Could anyone help me out with this problem? Am I on the right track? July 27th, 2017, 07:53 AM #2 Global Moderator   Joined: Dec 2006 Posts: 21,030 Thanks: 2260 $x = 1000/(1000/x^2)^2 = x^4/1000$, so $x = 10$. Similarly, $y = 10$, and so $z = 5$. Thanks from Country Boy and SenatorArmstrong July 27th, 2017, 08:16 AM   #3
Senior Member

Joined: Nov 2015
From: United States of America

Posts: 198
Thanks: 25

Math Focus: Calculus and Physics
Quote:
 Originally Posted by skipjack $x = 1000/(1000/x^2)^2 = x^4/1000$, so $x = 10$. Similarly, $y = 10$, and so $z = 5$.
Thanks a ton for clearing this up, sir! Tags area, minimize, surface, tank calculus

Click on a term to search for related topics.
 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 Omnipotent Calculus 4 December 20th, 2014 08:35 PM chuackl Calculus 2 December 27th, 2013 05:06 PM Survivornic Calculus 7 January 21st, 2013 07:52 AM rtseamar Calculus 1 January 24th, 2012 09:57 AM symmetry Algebra 5 February 3rd, 2007 06:09 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top      