A small pendulum is released in water. The maximum height that the pendulum reaches each time it swings decreases over time. Every $180$ seconds, the pendulum's maximum height is reduced by $90\%$ . If the pendulum has been swinging for $360$ seconds and now reaches a maximum height of $0.031$ centimeters, what was the maximum height of the pendulum, in centimeters, when it was initially released? $\newline$ ◻

Full solution

Q. A small pendulum is released in water. The maximum height that the pendulum reaches each time it swings decreases over time. Every

180

seconds, the pendulum's maximum height is reduced by

90\%

. If the pendulum has been swinging for

360

seconds and now reaches a maximum height of

0.031

centimeters, what was the maximum height of the pendulum, in centimeters, when it was initially released?

\newline

◻

Understand the problem: First, we need to understand the problem. The pendulum's maximum height decreases by $90\%$ every $180$ seconds. After $360$ seconds, the maximum height is $0.031$ centimeters. We need to find the initial maximum height.
Calculate retained height: Since the pendulum's height decreases by $90\%$ every $180$ seconds, it retains $10\%$ of its height after each interval. After $360$ seconds, which is two intervals, the pendulum would retain $10\%$ of its height twice.
Denote initial height: Let's denote the initial maximum height as $H$ . After the first $180$ seconds, the height would be $10\%$ of $H$ , which is $0.1H$ . After the second $180$ seconds, the height would be $10\%$ of $0.1H$ , which is $(0.1)^2 \times H$ .
Set up equation: We know that after $360$ seconds, the height is $0.031$ centimeters. So, we can set up the equation $(0.1)^2 \times H = 0.031$ .
Solve for initial height: Now, we solve for $H$ . $(0.1)^2$ is $0.01$ , so the equation becomes $0.01 \times H = 0.031$ . To find $H$ , we divide both sides of the equation by $0.01$ .
Solve for initial height: Now, we solve for $H$ . $(0.1)^2$ is $0.01$ , so the equation becomes $0.01 \times H = 0.031$ . To find $H$ , we divide both sides of the equation by $0.01$ . Dividing $0.031$ by $0.01$ gives us $H = \frac{0.031}{0.01} = 3.1$ centimeters.