Ladders on the walls

Guys please help me with this problem... I've been eating my head off but cannot solve this problem. I used similarity and tried finding the values of AD and BC bt im left with a bi-quadratic equation y^4 - 20y^3 + 700y^2 - 14000y + 70000 = 0 where y = BC Please help. Thank you

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Comments

Here's one approach. Note that:

$\frac{OE}{BC} = \frac{AE}{AB}, \ \frac{OE}{AD} = \frac{EB}{AB} \implies \frac{OE}{BC} + \frac{OE}{AD} = 1.$

If we denote $x=AB$ , then this gives $\frac{10}{\sqrt{30^2 - x^2}} + \frac{10}{\sqrt{40^2 - x^2}} = 1$ . You'll probably still get a quartic equation, but at least all terms are even powers of $x$ so you can substitute $u = x^2$ and solve there.

[ Edit: actually this doesn't seem to work since you still get a quartic equation in $u$ after expansion. ]

C Lim - 7 years, 9 months ago

There is no way to get a nice exact solution for this problem. Values were poorly chosen. Your equation is correct. The correct result is close to AB = 26.

They probably wanted AB = 24, AD = 32, BC = 18, but messed up by the choice of OE.

It is hard to choose 3 integer values for AC, BD and OE so that AB is integer. I can't find anything better than: BD = 119 AC = 105 OE = 30

Kelvin Azevedo Santos - 7 years, 9 months ago

According to what I have calculated, AB=28.5 m.

First we will use C Lim's approach.

OE/BC + OE/AD =1------------------(1)

OE/AD = OB/BD ---------------(2)

OE/BC = OA/AC ---------------(3)

Adding (2) and (3),

OE/AD + OE/BC = OB/BD + OA/AC

1 = OB/40 + OA/30 [From --------(1) and putting the values of BD and AC]

Solving further, we get an equation,

120=3(OB) +4(OA) --------------(4)

Now, ODA is similar to OBC, which gives us,

OD/OC = OB/OA

(BD-OB)/(AC-OA) = OB/OA

(40-OB)/(30-OA) = OB/OA

Solving further, we get, 4(OA) = 3(OB) --------------(5)

From (4) and (5), we get OA=15 m and OB=20 m.

Now, using pythagoras theorem in triangles OEA and OEB, we get

EA = 5 root5 m and EB = 10 root3 m.

AB = EA +EB = 5 x 2.236 + 10 x 1.732 = 28.5 m

Prakkash Manohar - 7 years, 3 months ago

you should have assumed the sides as integers.i know it's a guess but worth trying.if you assume them as integers then you can easily conclude that the answer is 24

Kishore Saldanha - 7 years, 9 months ago

how do u conclude? please elaborate a little. thank u

Sagnik Saha - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$