A ladder 20 feet long leans up against a house.
The bottom of the ladder starts to slip away from the house at 0.27 feet per second.
How fast is the tip of the ladder along the side of the house slipping when the ladder is 7 feet away from the house? (Round to 3 decimal places.)
\[
\frac{\mathrm{ft}}{\mathrm{sec}}
\]
Consider the angle the bottom of the ladder makes with the ground.
How fast is the angle changing (in radians) when the ladder is 7 feet away from the house?

Step 1 ：Let's denote the distance from the bottom of the ladder to the house as \(x\), the height of the ladder on the wall as \(y\), and the angle between the ladder and the ground as \(\theta\).

Step 2 ：According to the Pythagorean theorem, we have \(x^2 + y^2 = 20^2\).

Step 3 ：Differentiating both sides with respect to time \(t\), we get \(2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0\).

Step 4 ：Since the bottom of the ladder is slipping away from the house at 0.27 feet per second, we have \(\frac{dx}{dt} = 0.27\) feet per second.

Step 5 ：When the ladder is 7 feet away from the house, we can substitute \(x = 7\) and \(\frac{dx}{dt} = 0.27\) into the equation, and we get \(2*7*0.27 + 2y\frac{dy}{dt} = 0\).

Step 6 ：Solving for \(y\) using the Pythagorean theorem when \(x = 7\), we get \(y = \sqrt{20^2 - 7^2} = \sqrt{399} = 20\sqrt{0.9975}\).

Step 7 ：Substituting \(y = 20\sqrt{0.9975}\) into the equation, we get \(2*7*0.27 + 2*20\sqrt{0.9975}\frac{dy}{dt} = 0\).

Step 8 ：Solving for \(\frac{dy}{dt}\), we get \(\frac{dy}{dt} = -\frac{2*7*0.27}{2*20\sqrt{0.9975}} = -\frac{0.27}{\sqrt{0.9975}}\).

Step 9 ：Rounding to 3 decimal places, we get \(\frac{dy}{dt} = \boxed{-0.270}\) feet per second.

Step 10 ：Next, we consider the angle \(\theta\) the bottom of the ladder makes with the ground. According to the trigonometric function, we have \(\cos\theta = \frac{x}{20}\).

Step 11 ：Differentiating both sides with respect to time \(t\), we get \(-\sin\theta\frac{d\theta}{dt} = \frac{1}{20}\frac{dx}{dt}\).

Step 12 ：Substituting \(\frac{dx}{dt} = 0.27\), \(x = 7\), and \(\cos\theta = \frac{7}{20}\) into the equation, we get \(-\sqrt{1 - (\frac{7}{20})^2}\frac{d\theta}{dt} = \frac{1}{20}*0.27\).

Step 13 ：Solving for \(\frac{d\theta}{dt}\), we get \(\frac{d\theta}{dt} = -\frac{0.27}{20\sqrt{1 - (\frac{7}{20})^2}}\).

Step 14 ：Rounding to 3 decimal places, we get \(\frac{d\theta}{dt} = \boxed{-0.019}\) radians per second.