A fence 3 feet tall runs parallel to a tall building at a distance of 4 feet from the building...

Question

A fence 3 feet tall runs parallel to a tall building at a distance of 4 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

Here are some hints for finding a solution:

Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence.

If the ladder makes an angle 0.65 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence is 4.957 and the distance along the ladder from the top of the fence to the wall is 5.0244.

a) Using these hints write a function  which gives the total length of a ladder which touches the ground at an angle , touches the top of the fence and just reaches the wall. 

L(x) = [ ]

b) Use this function to find the length of the shortest ladder which will clear the fence.

The length of the shortest ladder is [ ] feet.

Answer 1

This problem is deceptively more difficult than it appears. The previous post assumes that the solution is a 45-45-90 degree triangle. That would be the case only when the fence height and the distance between the fence and the building were the same.

The correct solution is to use Similar Triangles and Calculus. Let x=the distance that the foot of the ladder is from the fence, and y=the height that the top of the ladder reaches on the building. The following equation is readily seen:

3 / x = y / (x+4)

from which we easily get

y = 3 (x + 4) / x, or

y = 3 (1 + 4/x)

Let L(x) = the length of the ladder as a function of x. Pythagoras' formula says:

L² = (x + 4)² + y²

= (x + 4)² + 9 (1 + 4/x)²

Take the derivative of both sides and set to zero:

2 L L' = 2 ( x + 4 ) + 18 (1 + 4/x) (-4/x²) = 0

After some algebra, we get the quartic:

x⁴ + 4 x³ - 36 x - 144 = 0

The only positive real root, found numerically using Octave-Online, is

x = 3.30193

y = 3(1+4/x) = 6.63424

and

L = 9.86566

By the way, as said above, the angle was not 45 degrees; it turned out to be 47.74 degrees.

Answer 2

Small triangle: cos(θ) = a/b

Big triangle: cos(θ) = (4 + a)/L

So a/b = (4 + a)/L

However, using the Pythagorean Theorem on the small triangle, a² + 3² = b² meaning b = √(a² + 9)

Plugging that in to the equation we got from the cosine, we get::

a/√(a² + 9) = (4 + a)/L

Cross multiplying:

L*a = (4 + a)√(a² + 9)

Then dividing by a, we get a function in terms of a:

L(a) = (4 + a)√(a² + 9)/a

Using Calculus, we can take the derivative and set it equal to zero to find the minimum. You don't really say if you are in Calculus or not so another way to find the minimum is to graph the function and find the minimum value. Graphing it we get something like this:

Where the minimum is at a = 3.3019, L = 9.866 making θ = 42.26°

Answer 3

The shortest ladder has to be at 45°.

The length of the ladder between the fence and the building is side "c".

Upper: a=4'; b=4', α=45°; β=45°; γ=90°.

Lower: a2=3'; b2=3'; α2=45; β2=45; γ2=90.

The length of the ladder side "c"=a^2+b^2=c^2=32^.5=5.6568............

"c2"= 18^.5=4.242.........

"c"+"c2"=9.899........ft

rad=180/pi.