ladder problem calc

Question

A fence 24 feet tall runs parallel to a tall building at a distance of 6 ft from the building as shown in the diagram.

We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

[A] First, find a formula for the length of the ladder in terms of θ

L(θ)=

Now, find the derivative, L'(θ)

L'(θ)=

Once you find the value of θthat makes L'(θ)=0, substitute that into your original function to find the length of the shortest ladder.

L(θmin)≈

Answer 1

Draw a figure!

There are 2 relevant similar right triangles.

One is outside the fence and the fence and has leg 24. The hypotenuse is 24 csc θ.

The other is between the fence and the wall and has leg 6. The hypotenuse is 6 sec θ.

The ladder length L is the sum of these 2 and the critical value will be where dL/dθ = 0.

I will leave the differentiation and algebra to you!