Find the associated half-life time or doubling time. (Round your answer to three significant digits.)
Q = 900e^(−0.025t)
th =

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).

---

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 \( e^x \).

Find the associated half-life time or doubling time. (Round your answer to three significant digits.) Q = 900e^(−0.025t) th =

Learn how to calculate the half-life for an exponential decay model using the equation Q = 900e^(-0.025t). This detailed step-by-step solution explains key concepts like exponential decay, the natural logarithm, and half-life formulas, suitable for students in Grades 10-12.

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