# Thread: Calculus Word Problem Ladder

1. ## Calculus Word Problem Ladder

A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall? Evaluate your answer numerically. (Round the answer to three decimal places.)

2. Originally Posted by kashifzaidi
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 6 ft from the wall? Evaluate your answer numerically. (Round the answer to three decimal places.)
make a sketch of the problem.

let $x$ = horizontal distance ... $\frac{dx}{dt} = 1.5$ ft/sec

let $\theta$ = angle

determine the trigonometric relationship equation using $x$ , $\theta$, and the 10 ft ladder.

take the time derivative, sub in your given/calculated values, then determine $\frac{d\theta}{dt}$

3. You want to find: $\frac{d\theta}{dt}=\frac{d\theta}{dx}\frac{dx}{dt}$

Note that:

$x=10cos(\theta)$

$\frac{dx}{d\theta}=-10sin(\theta)$

$\frac{d\theta}{dx}=\frac{-csc(\theta)}{10}$

I think that's the trick you're looking for. Just plug this into the equation for $\frac{d\theta}{dt}$.

To find the numerical answer, you can use the fact that you are given $6=10cos(\theta)$ at this point.