We are tasked with converting the given exponential function:

$f(t) = 10(0.989)^t$

into the form:

$f(t) = Q_0 e^{-kt}$

and rounding all coefficients to four significant digits.

Steps:

1. Start with the given function: $f(t) = 10(0.989)^t$

2. Rewrite $(0.989)^t$ in terms of the base $e$ using the property $a^t = e^{t \ln a}$: $(0.989)^t = e^{t \ln(0.989)}$

3. Simplify $t \ln(0.989)$: $\ln(0.989) \approx -0.011080$ (rounded to six decimal places for intermediate calculations).

   Thus: $(0.989)^t = e^{-0.01108t}$

4. Substitute this back into the function: $f(t) = 10 e^{-0.01108t}$

5. Identify the coefficients:

   • $Q_0 = 10$
   • $k = 0.01108$

Final Answer:

$f(t) = 10 e^{-0.01108t}$

Would you like a breakdown of how to handle such conversions in other contexts or any further clarifications?

Related Questions:

1. How do logarithmic properties help convert exponential functions?
2. What is the significance of the base $e$ in exponential functions?
3. How can you verify $\ln(0.989)$ using a calculator?
4. Why do we use four significant digits in scientific computations?
5. What are some applications of exponential decay models in real life?

Tip:

Always keep intermediate calculations with extra precision and round only the final result to the required number of significant digits.