Here is the problem I'm working on:

A ladder 20ft long leans against a house. If the foot of the ladder is moving away from the house at the rate of 2ft/s, find how fast the slope of the ladder is decreasing when the foot of the ladder is 12ft from the house.

Here is what I have so far:

Given:

- $\displaystyle \frac{dx}{dt}=2$
- x=12
- z=20
Find:

- $\displaystyle \frac{d\theta}{dt} $ when x = 12
My Work:

- By the theorem of Pythagoras:

- y=16
- $\displaystyle sin\theta = \frac{y}{z} = \frac{16}{20} = \frac{4}{5}$
- $\displaystyle cos\theta=\frac{x}{20}$
- $\displaystyle x=20cos\theta$
- $\displaystyle \frac{dx}{dt}$ = $\displaystyle 20(-sin\theta)*\frac{d\theta}{dt}$
- $\displaystyle 2 = 20*\frac{-4}{5}*\frac{d\theta}{dt}$
$\displaystyle \frac{d\theta}{dt} = \frac{-1}{8} rad/s$

Any tips on where I went wrong?

The book, however, came up with $\displaystyle \frac{25}{72} per second$

Thanks in advance!