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  • Semi - OT Trig Problem

    Semi - OT because this problem comes from the book 'Machine Shop Trade Secrets' page 272.

    "A ten-foot ladder is leaned up against a wall. A two-foot cubic box is placed under the ladder and up against the wall. The angle of the ladder is adjusted so that it is touching the floor, the wall, and the corner of the box. How high up the wall does the ladder touch?"

    The answer is given in the Appendix (9.6771 feet) but can anyone explain to me how to arrive at that answer?

  • #2
    Math was my greater weakness but that was in the days before affirmative action.

    Picturing this in my mind I can see a side view of a square and two triangles. If you know the heigth of the box (it's cubic as phrased in your question) can't you deduce that having it touch the ladder would in affect creat two triangles of which you know two sides of both of them and there by able to compute the third leg? Just asking, there's likely one of those letter answers but I won't be able to follow it LOL

    [This message has been edited by Your Old Dog (edited 06-15-2005).]
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    Thank you to our families of soldiers, many of whom have given so much more then the rest of us for the Freedom we enjoy.

    It is true, there is nothing free about freedom, don't be so quick to give it away.

    Comment


    • #3
      <font face="Verdana, Arial" size="2">Originally posted by crews1:
      Semi - OT because this problem comes from the book 'Machine Shop Trade Secrets' page 272.

      "A ten-foot ladder is leaned up against a wall. A two-foot cubic box is placed under the ladder and up against the wall. The angle of the ladder is adjusted so that it is touching the floor, the wall, and the corner of the box. How high up the wall does the ladder touch?"

      The answer is given in the Appendix (9.6771 feet) but can anyone explain to me how to arrive at that answer?
      </font>
      Two words Auto Cad.

      Lol okay now flame me math gurus.

      Comment


      • #4
        I'm working on it... I need more information to give you a correct answer.

        Looking at the side of the ladder, how wide are the ladder rails? Are they square at the bottom, or are they round? What about the top of the ladder. Is the top rounded, or square? (I.E: Is one corner of the ladder touching the ground, and one corner touching up at the top, or is the ladder pivoting at the bottom?

        All of these items affect the rise over run when leaning it.. If it's not specified, I'll assume the ladder has no width and just pivots on an infinitly small radius/point.

        -Adrian

        Comment


        • #5
          Sorry, but that was all the info that there was.
          I *assume* that it isn't a trick question and that the 'ladder' and 'box' are used to make it easier to visualize.
          He has a sketch that shows it as a triangle with the hypotenuse of 10 and the 2-foot square at the base.

          Comment


          • #6
            This one is a classic. Been used more times than I can count. Why? Read on.



            We are trying to solve for y in the diagram, the distance from the floor to the top of the ladder. We have two unknowns, x and y so we need two independent equations with these variables in order to solve for them.

            x^2 + y^2 = 10^2 Obvious Equation 1

            Looking at areas we can see that the area of the large triangle is xy/2 and the areas of the smaller triangles are 2(x-2)/2 and 2(y-2)/2 respectively. These quickly reduce to x-2 and y-2. And the area of the square is 2*2 or 4. Since the area of the large triangle is the sum of the areas of the two small triangles and the square, we can say:

            xy/2 = (y-2) + (x-2) + 4

            Reducing,

            xy = 2y - 4 + 2x - 4 + 8

            xy = 2y + 2x Equation 2

            Working with Equation 2,

            xy -2x = 2y

            x(y-2) = 2y

            x = 2y/(y-2) Equation 3

            Plugging the value of x in equation 3 into equation 1 we get

            (2y/(y-2))^2 + y^2 = 100

            4y^2/(y^2 -4y + 4) + y^2 = 100

            Now you have to multiply all terms by the denominator of the first term (y^2 -4y + 4)

            4y^2 + y^2(y^2 -4y + 4) = 100(y^2 -4y + 4)

            This reduces to:

            y^4 - 4y^3 - 92y^2 + 400y - 400 = 0 Equation 4

            In short, you have to solve a fourth degree equation for y. That's why it is a classic. Not many people can solve such an equation. Mathematicians can. I did in high school and college and I did this problem about three times in the past but I will pass on the rest now. It isn't fun. It gets really messy. You will get four answers because it is a fourth order equation. If I remember, two are immaginary numbers and the other two are for the ladder at a steep angle and a shallow angle (x and y are swapped).

            If anyone wants to complete it, have fun.

            Oh, I don't think I have ever seen any shortcuts for this process - except the CAD route which is approximate so in the world of math puzzles it doesn't count.

            Paul A.
            Paul A.

            Make it fit.
            You can't win and there is a penalty for trying!

            Comment


            • #7
              Thank you for taking the time to post that detailed explanation. I appreciate it.

              Comment


              • #8
                I'm inventing a shortcut right now. This might take me awhile.


                Comment


                • #9
                  Don't take this to your local trig teacher as I cannot verify without my books (can't find them). I am fairly certin you will have to use trigonometric identities to solve this one. You will have to express one of the unknowns as a relationship to the other known values and them solve algebraicly using the trig identities. I will do some research and post later.

                  Comment


                  • #10
                    Where's Klotz when ya need em?

                    He's probably reading this and laughing his ass off.

                    Comment


                    • #11
                      It's a quadratic equation.

                      See here:

                      http://mathcentral.uregina.ca/QQ/dat...98/blade1.html
                      Free software for calculating bolt circles and similar: Click Here

                      Comment


                      • #12
                        Crews1,

                        I found there were several ways to solve this problem. Google searching was not one of them

                        I came up my own equation to calculate the height of the ladder against the wall. This program lets you enter the length of the ladder, and the size of the box. The program calculates how high the ladder will reach on the wall when touching the floor, the box, and the wall at the same time.

                        My equation comes up with 9.66865 feet. Your result of 9.6771 is .008 taller. I wonder who's method is more accurate.

                        -Adrian


                        Code:
                        /*****************************************/
                        /*                                       */
                        /* Created 06-15-05: Adrian Michaud      */
                        /*                                       */
                        /* Ladder and Box Calculation Shortcut   */
                        /*                                       */
                        /*****************************************/
                        
                        #include &lt;math.h&gt;
                        
                        void main(int argc, char **argv)
                        {
                        double ladder,box;
                        	  
                           if (argc &lt; 3)
                              {
                              printf("Usage: ladder height box");
                              exit(1);
                              }
                        
                           sscanf(argv[1], "%lf", &ladder);  
                           sscanf(argv[2], "%lf", &box); 
                        
                           printf("Height of Ladder   : %lf feet\n", ladder);
                           printf("Size of Box        : %lf feet\n", box);
                        
                           printf("Height of Ladder is: %lg feet\n", 
                              cos(atan(box/((ladder-box)-
                             ((sin(atan(box/(ladder-box)))
                               *ladder)-box))))*ladder);
                        }
                        
                        
                        Z:\&gt;cl ladder.c
                        Microsoft (R) 32-bit C/C++ Optimizing Compiler Version 12.00.8804 for 80x86
                        Copyright (C) Microsoft Corp 1984-1998. All rights reserved.
                        
                        ladder.c
                        Microsoft (R) Incremental Linker Version 6.00.8447
                        Copyright (C) Microsoft Corp 1992-1998. All rights reserved.
                        
                        /out:ladder.exe
                        ladder.obj
                        
                        Z:\&gt;ladder
                        Usage: ladder height box
                        Z:\&gt;ladder 10 2
                        Height of Ladder   : 10.000000 feet
                        Size of Box        : 2.000000 feet
                        Height of Ladder is: 9.66865 feet
                        Z:\&gt;
                        
                        [[email protected] dev]$ gcc ladder.c -oladder-linux -lm
                        [[email protected] dev]$ ./ladder-linux 10 2
                        Height of Ladder   : 10.000000 feet
                        Size of Box        : 2.000000 feet
                        Height of Ladder is: 9.66865 feet
                        [[email protected] dev]$

                        Source Code Link:

                        http://www.bbssystem.com/ladder/ladder.c

                        WIN32 .EXE:

                        http://www.bbssystem.com/ladder/ladder.exe

                        Linux X86 Binary executable:

                        http://www.bbssystem.com/ladder/ladder-linux

                        -Adrian

                        Comment


                        • #13
                          Link is easier than writing all that stuff here.

                          I have a question for you Adrian. What are the maximum possible internal angles of an equilateral triangle in spherical trig?

                          [This message has been edited by Evan (edited 06-16-2005).]
                          Free software for calculating bolt circles and similar: Click Here

                          Comment


                          • #14

                            Sorry folks!

                            Read the question.

                            What happens when the ladder falls!

                            There are 2 answers.

                            Norman

                            Comment


                            • #15
                              No Evan, it is still a fourth order equation that just happens to be simple enough to be solved by use of the quadratic formula. It will still have four solutions, two of which will be physicially impossible and are thus discarded. These four solutions are listed in the last line.

                              But I guess it does qualify as a simplification. Although, not many will be clever enough to think of that trick. You only found it with a web search.

                              Paul A.
                              Paul A.

                              Make it fit.
                              You can't win and there is a penalty for trying!

                              Comment

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