
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 30th, 2012, 11:43 PM  #1 
Senior Member Joined: Apr 2008 Posts: 194 Thanks: 3  my correct solution but another solution?
Let's consider the following optimization problem. This is how I solve it. Find the length of the shortest ladder that can extend from a vertical wall, over a fence 2 metres high located 1 metre away from the wall, to a point on the ground outside the fence. my solution Let x be the angle of inclination of the ladder from the ground. Using two rightangled triangles, we obtain the length L of the ladder as a function of x. L = 1/cos(x) + 2/sin(x) differentiating L with respect to x and setting L' to zero give 0 = sin(x)/(cos(x))^2  2cos(x)/(sin(x))^2 It follows that (tan(x))^3 = 2 Next, I use the identity (sec(x))^2 = 1 + (tan(x))^2 and the relationship sin(x) = tan(x)*cos(x) to find the answer. L = (1 + 2^(2/3))^(3/2) which is approximately equal to 4.16. I know my solution is correct. I am wondering if there is another way to solve the problem without using trigonometry. Can someone help me? Thanks. 
January 31st, 2012, 02:48 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,919 Thanks: 2203 
[attachment=0:1opi6z83]ladder.gif[/attachment:1opi6z83] In the above diagram, OBCA is a rectangle, EA = t metres and the ladder AB has length ((t + 1)/t)?(4 + t²) metres. For t > 0, it's straightforward to use calculus to show that AB² has a minimum when t = 2^(2/3). As the ladder is of minimum length, CF? AB. It can be deduced that t = 2^(2/3) without using calculus. Either approach confirms (without use of trigonometry) that your solution is correct. 

Tags 
correct, solution 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Units of this ring: Z?[x]/(x² + 1)?Is my solution correct?  ricsi046  Abstract Algebra  6  March 17th, 2014 10:14 AM 
Is this solution correct?  rain  Calculus  1  November 19th, 2013 08:22 AM 
Is this correct solution?  rain  Calculus  4  October 12th, 2013 01:08 PM 
Find correct system for a specific solution  weekStudent  Linear Algebra  1  January 10th, 2013 12:10 PM 
Is my solution correct? (Related rates)  SarahT  Calculus  1  March 6th, 2012 11:29 AM 