Solving the Crossed Ladder Problem

[kingredg]

my teacher gave my class this problem earlier this week
and we need to know it by the end of next week

anyone knows how to solve this


sorry for the bad drawing

 

[slider142]

First, you can use the obvious Pythagorean theorem to get two equations in 3 unknowns. Get the third equation by using CF and the ratios of the two similar triangles to their larger counterparts.

 

[kingredg]

^^^how can i use the Pythagorean theorem when i only have 1 side length for both

and by the way I'm only a freshman

what's the CF?

 

[kamerling]

if you guess that AE = x. you can calculate AB and DE with pythagoras, and then
you have CF/AF = DE/AE because the triangles AED and AFC are similar. That gets you
AF, and you can find EF in the same way. Finally you must have AE = AF + EF.
I believe you'll get a 4th degree equation in x^2 after two rounds of squaring and collecting terms. I'm going to solve this numerically.

 

[kingredg]

^^^i'll try that

 

[janhaa]

Is AE $$\;\; \approx \;\,26\,\text meters\,?$$

 

[kamerling]

I get the solution of $$\frac {CF}{\sqrt{{AD}^2 - x^2}} + \frac {CF}{\sqrt{{BE}^2 - x^2}} = 1$$

wich is 26.0328775442...

 

[kingredg]

^^^sorry for asking such a dumb question but
how did you find the X to solve it

 

[kamerling]

^^^sorry for asking such a dumb question but
how did you find the X to solve it

I just set AE equal to x. this gives you

DE = sqrt(AD*AD - x*x) using pythagoras in the triangle DEA
AF = CF * x / DE using the fact that CF/AF = DE/AE because the triangles AED and AFC are similar.

combined this gives $$AF = \frac {x(CF)}{\sqrt{{AD}^2 - x^2}}$$

you can do the same on the other side to get $$EF = \frac {x (CF)}{\sqrt{{BE}^2 - x^2}}$$

Finally you must have AF + EF = AE = x

Substituting the previous expressions for AF and EF in this and dividing by x gives:

$$\frac {CF}{\sqrt{{AD}^2 - x^2}} + \frac {CF}{\sqrt{{BE}^2 - x^2}} - 1 = 0$$

The easiest way to solve this is to type 10/sqrt(1600-x^2)+10/sqrt(900-x^2)-1
into this webpage http://wims.unice.fr/wims/wims.cgi?session=6B26C0C5C3.3&+lang=en&+module=tool%2Fanalysis%2Ffunction.en"

 

[kingredg]

I just set AE equal to x. this gives you

DE = sqrt(AD*AD - x*x) using pythagoras in the triangle DEA
AF = CF * x / DE using the fact that CF/AF = DE/AE because the triangles AED and AFC are similar.

combined this gives $$AF = \frac {x(CF)}{\sqrt{{AD}^2 - x^2}}$$

you can do the same on the other side to get $$EF = \frac {x (CF)}{\sqrt{{BE}^2 - x^2}}$$

Finally you must have AF + EF = AE = x

Substituting the previous expressions for AF and EF in this and dividing by x gives:

$$\frac {CF}{\sqrt{{AD}^2 - x^2}} + \frac {CF}{\sqrt{{BE}^2 - x^2}} - 1 = 0$$

The easiest way to solve this is to type 10/sqrt(1600-x^2)+10/sqrt(900-x^2)-1
into this webpage http://wims.unice.fr/wims/wims.cgi?session=6B26C0C5C3.3&+lang=en&+module=tool%2Fanalysis%2Ffunction.en"

i try plugging that into my calculator is it okay if i just press the x (variable) or do i have to have in a specific number