ladder

[chilliman]

A 3 foot cube of concrete is flush against a perpendicular wall. Leaning against the wall is a 30 foot long ladder which is just touching the block. How high up the wall does the ladder touch?

The answer is easy to work out with graph paper but the geometric solution has eluded me for many months.
Can anyone provide this geometric calculation

[jan1956]

In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Pythagoras' Theorem

so 30 x 30 = 3 x 3 + X
900 = 9+X
X = the square root of 891
=29.849623 ft
so 29 ft 10inches

[jan1956]

not sure how old you are chiliman
but this is (or was when I went to school) basic geometry
http://www.mathsisfun.com/pythagoras.html
that is the page for it all
jan

[mjd]

jan1956, remember the old adage, always read the question at least twice. The bottom of the ladder is not 3ft out from the wall but the ladder is touching, the ground, the upper outer corner of the block and the wall

[jan1956]

right sorry chiliman for giving the wrong answer

[mjd]

let the top of the ladder be xft from the top of the block, and let the bottom of the ladder be y ft from the block. that is the ladder is 3+x up the wall and 3+y out from it.
By similar triangles x/3 = 3/y therefore xy = 9

Now (x+3)**2 + (y+3)**2 = 30**2 = 900, multiplying out
x**2+6x+9+y**2+6y+9 = 900 , substitute xy for 9 and collecting terms
x**2 + 2xy+y**2 + 6(x+y) =900
(x+y)**2 + 6(x+y) - 900 =0
Solving for (x+y) gives x+y = 27.15
substituting 9/x for y and multiplying through & rearranging gives x**2 - 27.15x + 9 = 0 giving x= 26.81
and the top of the ladder 29.91 feet up the wall

[chilliman]

Thanks for your thinking folks.
If I was really pedantic, jan1956 I would tell you to put brackets around the 3s. Remember BODMAS.
My age will remain a mystery!

[jan1956]

if I were doing it for an exam I would have done, if you had come back with the bracket answer instead of a thanks I would have thought you very rude, but as I gave the wrong answer will let that pass.
hope you are not checking this for spelling too
jan

[TeeGee]

Jan1965 was quite correct in omitting the brackets round the 3's as multiplication takes priority over addition.
The brackets therefore would have been superfluous.

[chilliman]

Thanks again for the posts.
Nearly kick off so I'm rushing now.
MJD, I'm not sure of the "similar triangles" theory but am working through the rest of the algebra.
You obviously spent time on this for which I give you many thanks.

[mjd]

The large triangle on the top of the block is "similar" to the small triangle at the side of the block, and all that this means is that one is an enlargement of the other, therefore all ratios are equal so x/3 = 3/y.

[chilliman]

Thanks once again MJD. The tangents kicked in just after kick off.

[Catron]

Is it just me or does there appear something wrong here? As jan1956 says, if the bottom of the ladder WAS three feet way from the wall, the top would be 29.849623 feet up the wall.

Now, if the bottom of the ladder is further away from the wall, as it has to be to be touching the concrete cube, it follows that the top of the ladder has to be lower down the wall than 29.849623 feet. Yet mjd gives the answer as 29.91 feet (ie further UP the wall).

I am not clever enough to follow mjd's workings so can they or somebody else tell me where my logic is going astray please?

[chilliman]

mjd did make a slight mistake with his maths at the end.
It should have been 26.81+3 which of course = 29.81. Well spotted

[mjd]

Chilliman
I was so pleased with getting to the end that didn't check my answer. Slapped wrist for me.

Does your username imply a liking for chillies? I only ask because I've just been to the Peppers by Post open day & have bought some Dorset Nage chillies which are reputed to be the hottest ever. I now need to find a suitable recipe.

[chilliman]

Hello again mjd.
Yes, I'm what they call a "chilehead". I've seen and heard of Dorset Naga but not yet tried. The previous hottest was the Red Savina which I've grown a few times and do leave an afterburn.
Take a look at my website www.chilligrower.co.uk
As for recipes, there are millions on the web, most of which are good. Enjoy.