FInding feet of a wall

Question

A fence 6 feet tall runs parallel to a tall building at a distance of 5 feet from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

leave answer in feet.

Answer 1

Hello,

The quick answer is approx. 15.47 ft (video details).

Thank you,

Dan

Answer 2

I got an answer slightly different from Yefim's, as follows:

The ladder is modelled by the equation y=6x/a, where a is the distance from the foot of the ladder to the wall.

This means the co-ordinates of the point where the top of the ladder touches the wall are

[(a+5), 6(a+5)/a]

The Pythagorean theorem gives the expression for L in terms of a.

Differentiating that expression with respect to a gives a=180^1/3 and L=15.534880554.

Answer 3

Let θ is angle between ladder and ground. Then L = 5/cosθ + 6/sinθ; dL/dθ = 5sinθ/cos²θ - 6cosθsin²θ = 0;

5sin³θ - 6cos³θ = 0; tanθ = (1.2)^1/3; θ = tan^-1(1.2^1/3).= 46.74°

L_min = 5/cos46.74° + 6/sin46.74° = 15.621 feet.

There is no maximum because if θ→0 then L →∞