Optimization

Hioj · 2010-02-04 08:10:14

NB: This is translated from a different language to English, but I hope you will be able to understand it anyway.

Assignment
A fence must be put up by a house as shown in the attached figure. The house is the hatched and blue part, and the fence is indicated by the broken lines. In total 60 m fence must be put up (i.e. |AB|+|BC+|CD|+|DE|=60), and all angles in the fence are right angles.

The length of line AB is designated x.
1) Determine the lengths |CD|, |BC| and |DE| expressed by x.
2) Determine the area of the fenced part, expressed by x.
3) Determine the dimensions of the fence, so that the area of the fenced part become as big as possible.

Solution
1) It's obvious that |CD| is

. I can't figure out how to find |BC| and |ED| expressed by x. So this is where I get lost.
2) Expressing the area by x is .
3) How do I make a function to describe this?

Thanks in advance!

Figure:

John E. Franklin · 2010-02-04 10:54:26

AB = x
( I didn't do it that way though)

Last edited by John E. Franklin (2010-02-05 07:41:36)

Hioj · 2010-02-05 06:55:11

Thank you so much for replying. Can you please elaborate on how you got that as a result? (i.e. how you solved part 1, 2 and 3)

John E. Franklin · 2010-02-05 07:40:50

I didn't use the x = AB method, so do you really want my way?
I used BC and CD in my equations...

(2BC - 2) + (2CD - 8) = 60
2BC + 2CD = 70
BC + CD = 35

Area = BC times CD minus 6, where 6 is 2 times 3.
Area = (BC)(CD) - 6
And CD = 35 - BC, so put in (substitute)
Area = (BC)(35 - BC) - 6
Expand
Area = 35BC - BC^2 - 6
Use Calculus for downward parabola for maxima:
Aprime = A' = A dy/dx = 35 - 2BC
Set A dy/dx = A' = 0 for maxima:
0 = 35 - 2BC
2BC = 35
BC = 17.5
CD = 17.5
Hey, 17.5 meters both sides, made a mistake before...

soroban · 2010-02-05 08:03:35

.

                  *-------------*
                  |:::::::::::::|
            x    A|:::::::::::::|
    B o - - - - - o-------*:::::|
      |               5  2|:::::|
      |                   *-----o E
  y+2 |                      3  |
      |                         | y
      |                         |
    C o - - - - - - - - - - - - o D
                 x+8

                  *-------------*
                  |:::::::::::::|
          7.5    A|:::::::::::::|
    B o - - - - - o-------*:::::|
      |               5  2|:::::|
      |                   *-----o E
 19.5 |                      3  |
      |                         | 17.5
      |                         |
    C o - - - - - - - - - - - - o D
                15.5

John E. Franklin · 2010-02-05 10:45:33

Unfortunately, 17.5 times 17.5 minus 6 is greater than Soroban's 19.5 times 15.5 minus 6.
I don't know why, but it is, so find the error is Soroban's post somewhere...

John E. Franklin · 2010-02-05 11:02:14

I found where Soroban went wrong. The 23 - x should be 27 - x, once he had the 25 -x, you add 2, not subtract 2.
I hope so anyway...

John E. Franklin · 2010-02-05 11:10:30

Yes!! It was a simple arithmetic error.
DE = 25 - x
so BC = 27 - x
And if you work it through, then
Area = (x + 8 )(27 - x) - 6
27x - 8x + 27(8) -x^2 - 6 = Area
27x - 8x + 210 - x^2 = Area
19x + 210 - x^2 = Area
19 - 2x = Area
19 = 2x
x = 9.5
CD = x + 8 = 17.5 like John's answer, Yeah!! So Soroban did it right way, just a tiny number mistake!!
BC = 27 - x = 27 - 9.5 = 17.5 like John's answer, Yeah, soroban did it the right way!! Just a typo.

John E. Franklin · 2010-02-05 11:12:48

DE = 15.5 because it is 2 less BC.
AB = 17.5 - 8 = 9.5 meters.

Math Is Fun Forum

#1 2010-02-04 08:10:14

Optimization

#2 2010-02-04 10:54:26

Re: Optimization

#3 2010-02-05 06:55:11

Re: Optimization

#4 2010-02-05 07:40:50

Re: Optimization

#5 2010-02-05 08:03:35

Re: Optimization

#6 2010-02-05 10:45:33

Re: Optimization

#7 2010-02-05 11:02:14

Re: Optimization

#8 2010-02-05 11:10:30

Re: Optimization

#9 2010-02-05 11:12:48

Re: Optimization

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