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? ◻

Question

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?  ◻

Bytelearn · Accepted Answer

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 * 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 * H = 0.031.Solve for initial height: Now, we solve for H. (0.1)^2 is 0.01, so the equation becomes 0.01 * 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 = 0.031 / 0.01 = 3.1 centimeters.

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