Question

A ladder 18 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when θ = π 3 ?

Answer 1

Answer:
`9\ ft/rad`

Explanation:

Given

Length of the ladder is
`l=18\ ft`

Angle between the wall and the ladder is
`\theta`

from the figure, we can write

`\Rightarrow \sin \theta=(x)/(18)\\\\\Rightarrow x=18\sin \theta`

Differentiate the above equation w.r.t
`\theta`

`\Rightarrow (dx)/(d\theta)=18\cos \theta\\\\\text{at }\theta=(\pi )/(3)\\\\\Rightarrow (dx)/(d\theta)=18\cos((\pi)/(3))\\\\\Rightarrow (dx)/(d\theta)=18* 0.5\\\\\Rightarrow (dx)/(d\theta)=9\ ft/rad`