(Calculus) A fence 3 feet tall runs parallel to a tall building at a distance of 4 feet from the building...

Question

A fence 3 feet tall runs parallel to a tall building at a distance of 4 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building.

Here are some hints for finding a solution:

Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence.

1)If the ladder makes an angle 0.65 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence.

2)The distance along the ladder from the top of the fence to the wall is

3)Using these hints write a function L(x)  which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just reaches the wall.

L(x)= 

4)Use this function to find the length of the shortest ladder which will clear the fence.

The length of the shortest ladder is  feet

Answer 1

Ladder to fence=a/sinα=c (rt triangle).

3/37.24sin=4.957'

Another rt triangle is the rest of the ladder and the building; b=4, β=52.76.

You want c=b/sinβ=4/sin52.76=5.0244......ft.

Building ht from ladder=a^2+b^2=c^2. But first you need the first triangle side b. b/sin52.76=c so b=3.946.....Add 4' to b (above) for the building triangle (ground).

Add the 2 ladder lengths. Reverse and use the formula above.