# A fence 4 feet tall runs parallel to a tall building at a distance of 2 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?

##### 2 Answers

8.32ft

#### Explanation:

Intending to use a **Lagrange Multiplier** , we are minimising

This is subject to constraint which comes from similar triangles that

so

using the constraint

ignoring the trivial solution we have

so ladder length

plot confirms authenticity of solution

The "proof" that this is a minimum comes from physical arguments. It is easy to imagine a ladder that has its base a distance

then

and

#### Explanation:

Fence height =

Fence distance =

Ladder length =

Solving for

Here

or

and