Question

A fence 16 feet tall runs parallel to a tall building at a distance of 4 ft from the building as shown in the diagram.

LADDER
16 ft
4 ft
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 0. (Hint: split the ladder into 2 parts.) Type theta for 0.
L(0)
[B] Now, find the derivative, L'(0).
Type theta for 0.
L'(0)
[C] Once you find the value of that makes L'(0) = 0, substitute that into your original function to find the length of the shortest ladder. (Give your answer accurate to 5 decimal places.

Answer 1

Explanation:

Let L be the length of the ladder.

sin theta = 16/ x

cos theta = 4/y

x = 16/ sin theta

y = 4 / cos theta

L = x + y = 16/sin theta + 4 / cos theta

(A) L = 16 cosec theta + 4 sec theta

(B) L' = -16 cosec theta cot theta + 4 sec theta tan theta = 0

Drawing a graph of this gives theta = 1.009

(C)

So the length of the shortest ladder

= 16/sin 1.009 + 4/cos 1.009

= 26.41465 ft.