roblem 2 . Because of air resistance, the length of each \( s v \) ing of a certain pendulum is \( 95 \% \) of the ngth of the previous swing. If the first swing has a length of 40 cm , find the total length the pendalum ill swing before coming to rest.

To find the total length the pendulum will swing before coming to rest, we can model the lengths of the swings as a geometric series. 1. **Identify the first term and the common ratio:** - The first swing length \( a = 40 \) cm. - Each subsequent swing is \( 95\% \) of the previous swing, so the common ratio \( r = 0.95 \). 2. **Sum of an infinite geometric series:** The formula for the sum \( S \) of an infinite geometric series is given by: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. 3. **Substituting the values:** - \( a = 40 \) cm - \( r = 0.95 \) Now, substituting these values into the formula: \[ S = \frac{40}{1 - 0.95} = \frac{40}{0.05} = 800 \text{ cm} \] Thus, the total length the pendulum will swing before coming to rest is \( \boxed{800} \) cm.