The equation $Q = 900e^{-0.025t}$ describes an exponential decay model, where $Q$ is the quantity at time $t$, and the decay rate is $0.025$ per unit time. To find the half-life $t_h$, we use the definition of half-life, which is the time it takes for the initial quantity to be reduced by half.

We can set $Q = \frac{900}{2} = 450$, since half of 900 is 450, and solve for $t$ in the equation:

$450 = 900e^{-0.025t}$

Step-by-step solution:

1. Divide both sides by 900 to isolate the exponential term: $\frac{450}{900} = e^{-0.025t}$ $0.5 = e^{-0.025t}$

2. Take the natural logarithm of both sides to get rid of the exponential: $\ln(0.5) = -0.025t$

3. Solve for $t$: $t = \frac{\ln(0.5)}{-0.025}$

   We know that $\ln(0.5) = -0.6931$, so: $t = \frac{-0.6931}{-0.025} = 27.724$

Thus, the half-life is approximately: $t_h \approx 27.7 \, \text{time units}$

Summary:

The half-life time $t_h$ is 27.7 (rounded to three significant digits).

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Do you want further details or have any questions?

Here are 5 follow-up questions to deepen your understanding:

1. How would the formula change if the decay rate were doubled?
2. What is the general formula for finding the half-life in an exponential decay model?
3. How does the initial quantity affect the half-life in this case?
4. Can you derive the doubling time from the same exponential model?
5. What is the significance of the decay constant in real-world applications?

Tip: When dealing with exponential models, remember that the natural logarithm (ln) is essential for solving equations involving $e^x$.