Length of Ladder?
Answer:
The shortest ladder would be 14ft 2in
The ladder would be inclined 45° towards the wall. It would stand on the ground 10ft from the wall, this is 4 ft from the fence. It would lean at the wall 10 ft above ground.
10ft * SQRT(2) = 14.14ft
This is 14ft 2in
(rounded to the next inch because USA is the only nation that was unable to convert to the international metric system for everyday applications)
approximately 18 feet
So...
>>>>>>I *~~~< ladder (xft)~~|<<<<<<<<<<<<<<<<<<<<<<...
>Fence I~~~~~*~~~~~~~~~~I Building<<<<<<<<<<<<<<<<<
>>>(4ft)I~~~~~~~~~~*~~~~~I wall<<<<<<<<<<<<<<<<<<<<
>>>>>>I_________________*I<<<<...
>>>>>>>>>>Ground(6ft)<<<<<<<<<...
(Connected Dotted stars is ladder)
So the length of the ladder is what you're trying to find? In that case...
Use the pythagorean theorem:
4squared+6squared=xsquared
16 + 36 =xsquared
52 = x squared
square root 52 = x
x = 7.211 ft
You can answer this question using Cartesian coordinate calculus but you end up with an order 4 polynomial that can only be solved graphically so a 2 decimal place answer is all you will get. An alternative approach is to use Polar Co-ordinate calculus which I shall now endeavor to do:
OK, draw yourself a diagram on a set of axis. Make the origin the top of the wall.
Now r(1) can be the length of ladder from origin to wall. r(2) is the ladder to the ground. We are using vectors so note that r(2) is pointing in the opposite direction to r(1).
r = total ladder length = r(1) - r(2)
The negative sign is because of the of opposite directions of r(1) and r(2).
Lets define r(1) and r(2) from what we know in the question. we will also call angle (x) the angle from ground to wall.
r(1) Cos (x) = 6
r(1) = 6 / Cos(x)
r(2) = -4/Sin(x)
r = r(1) - r(2)
r = 6/Cos(x) + 4/Sin(x)
Note that the negative sign now disappears.
We now need to differentiate this dr/dx
I will assume that you know how to differentiate this but if you don't then a full explanation can be obtained if you e-mail me.
Suffice to say that:
dr/dx
= [(6 Sin(x)) / (Cos (x))^2] - [(4 Cos(x)) / (Sin (x))^2
Now this looks complicated but don't worry, we can cross multiply to get a common denominator.
If we then make dr/dx = 0 we can multiply that common denominator across and it vanishes leaving us with:
6 (sin (x))^3 - 4 (cos (x))^3 = 0
So:
6 (Sin(x))^3 = 4 (cos (x))^3
Sin (x)^3 Cos (x) ^-3 = 4/6
Now remember that Sin / Cos = Tan
Cube root each side:
Tan (x) = Cube Root (2/3)
x = 41.13982782 degrees
Now that you know the angle that the ladder is placed at you should be able to calculate the length but in case you are still finding difficulties then:
r(1) = 7.967
r(2) = 6.080
r = 14.05 feet
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