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

Problem Understanding: Understand the problem. We are given that the maximum height of a pendulum decreases by 90% every 180 seconds. After 360 seconds, the maximum height is 0.031 centimeters. We need to find the initial maximum height of the pendulum.Number of Intervals: Determine the number of decrease intervals. Since the height decreases every 180 seconds and the pendulum has been swinging for 360 seconds, there have been 2 intervals of decrease. 360 seconds / 180 seconds per interval = 2 intervalsFirst Decrease Calculation: Calculate the height after the first decrease. Let's denote the initial maximum height as H. After the first 180 seconds, the height would be reduced by 90%, which means the pendulum retains 10% of its height. Height after first decrease = H * 10%Second Decrease Calculation: Calculate the height after the second decrease. The height after the first decrease is then subjected to another 90% decrease after the next 180 seconds. So, it retains 10% of the height from Step 3. Height after second decrease = (H * 10%) * 10%Equation Setup: Set up the equation with the known final height. We know that after the second decrease, the height is 0.031 centimeters. (H * 10%) * 10% = 0.031 centimetersSolving for Initial Height: Solve for the initial height H. To find H, we need to divide the final height by the product of the percentages from each interval. H = 0.031 centimeters / (10% * 10%) H = 0.031 centimeters / (0.1 * 0.1) H = 0.031 centimeters / 0.01 H = 3.1 centimeters

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