From the given problem, we aim to rewrite the equation $Q = \frac{1}{7} \cdot \sqrt[3]{6^t}$ in the form $Q = a e^{k t}$, then determine the growth factor $b$, starting value $a$, growth rate $r$, and $k$.

Step 1: Simplify the equation

The given equation is: $Q = \frac{1}{7} \cdot \sqrt[3]{6^t}$ We can rewrite $\sqrt[3]{6^t}$ as: $\sqrt[3]{6^t} = (6^t)^{1/3} = 6^{t/3}$ So, the equation becomes: $Q = \frac{1}{7} \cdot 6^{t/3}$

Step 2: Rewrite in exponential form

Using the property of exponents, $6^{t/3}$ can be rewritten as $e^{t \ln(6^{1/3})}$. Simplifying $\ln(6^{1/3})$, we get: $\ln(6^{1/3}) = \frac{1}{3} \ln(6)$ Thus: $Q = \frac{1}{7} \cdot e^{t \cdot (\ln(6)/3)}$

Step 3: Identify the components

Now the equation is in the form $Q = a e^{k t}$, where:

• $a = \frac{1}{7}$ (starting value)
• $k = \frac{\ln(6)}{3}$

The growth factor $b$ is given by $b = e^k$, and the growth rate $r$ is $r = k$.

Step 4: Calculate specific values

• $a = \frac{1}{7}$
• $k = \frac{\ln(6)}{3}$
  • Approximate $\ln(6) \approx 1.7918$, so $k = \frac{1.7918}{3} \approx 0.5973$
• $b = e^k = e^{0.5973} \approx 1.817$
• $r = k = 0.5973$

Final Answers:

1. $b = 1.817$
2. $a = \frac{1}{7}$ or approximately $0.1429$
3. $r = 0.5973$
4. $k = 0.5973$

Would you like further clarification or a step-by-step breakdown of any part?

Here are five related questions to deepen understanding:

1. How does the base $e$ relate to exponential growth and decay models?
2. Why is $\ln(6)$ used in converting the base-6 growth to base $e$?
3. How does the growth factor $b$ relate to the growth rate $r$?
4. Can you identify whether this is growth or decay based on the value of $r$?
5. How do the properties of logarithms simplify converting exponential equations to different bases?

Tip: When working with exponential functions, understanding logarithmic transformations is crucial for rewriting expressions in terms of $e$, the natural base.