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?

Answer 1

The initial maximum height of the pendulum, when released, was approximately 3.1 centimeters. Over 360 seconds, its height decreased exponentially by 90% every 180 seconds, reaching 0.031 centimeters.

The reduction in maximum height over time follows an exponential decay model. The formula for exponential decay is given by:

`\[ H(t) = H_0 * (1 - r)^{(t)/(T)} \]`

where:

-
`\( H(t) \)` is the height at time t,

-
`\( H_0 \)` is the initial height,

- r is the decay rate per unit time,

- T is the time it takes for the height to decrease by a factor of (1 - r).

In this case, the pendulum's maximum height decreases by 90% every 180 seconds, so r = 0.9 and T = 180 seconds.

Given that the pendulum has been swinging for 360 seconds, we want to find the initial height
`\( H_0 \)` when the current height H(t) is 0.031 centimeters.

`\[ 0.031 = H_0 * (1 - 0.9)^{(360)/(180)} \]\[ 0.031 = H_0 * (0.1)^2 \]`

Now, solve for
`\( H_0 \)`:

`\[ H_0 = (0.031)/(0.01) \]\[ H_0 = 3.1`

Therefore, the initial maximum height of the pendulum when it was released was 3.1 centimeters.