Optimization Problem

Question

A fence 10 feet tall runs parallel to a tall building at a distance of 2 ft from a building.

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 _____?

[B] Now, find the derivative, L'(θ) _____?

..

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

Answer 1

Let L be the length of the ladder. Let y be the length of the ladder from the building to the top of the fence.

Let z be the length of the ladder from the top of the fence to the ground. Let θ be the angle of the ladder

from the ground to the ladder. This is the same angle for the similar triangle for the ladder from the top of the fence leaning against the building.

L = y + z

cosθ = 2/y --> y = 2/cosθ

sinθ = 10/z --> z = 10/sinθ

1) L(θ) = 2/cosθ + 10/sinθ

2) L'(θ) = 2secθtanθ - 10cscθcotθ = 2sinθ/cos²θ - 10cosθ/sin²θ

3) 2sinθ/cos²θ - 10cosθ/sin²θ = 0

(2sin³θ-10cos³θ)/(sin²θcos²θ)

2sin³θ = 10cos³θ

tan³θ=5

θ = tan^-1(5^1/3) ≈ 60°

L(60) = 2/cos(60) + 10/sin(60) ≈ 4+11.5 = 15.5 feet

Answer 2

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