$
y(t) = 60000 e^{\frac{t}{9} \ln\left(\frac{5}{6}\right)}
$

To analyze the given function:

\[
y(t) = 60000 e^{\frac{t}{9} \ln\left(\frac{5}{6}\right)}
\]

we can rewrite it in a more interpretable form and understand its behavior over time.

### Step-by-Step Simplification:

1. **Exponent Properties**:
   The exponent has a logarithm, so let's simplify using the properties of exponents and logarithms:

   \[
   y(t) = 60000 \cdot e^{\ln\left(\frac{5}{6}\right) \cdot \frac{t}{9}}
   \]

2. **Simplify Exponential Form**:
   Using the identity $e^{a \cdot b} = (e^a)^b$, we can rewrite the expression as:

   \[
   y(t) = 60000 \cdot \left(e^{\ln\left(\frac{5}{6}\right)}\right)^{\frac{t}{9}}
   \]

   Since $e^{\ln(x)} = x$, this becomes:

   \[
   y(t) = 60000 \cdot \left(\frac{5}{6}\right)^{\frac{t}{9}}
   \]

3. **Interpretation**:
   This is an exponential decay function. The base $\frac{5}{6} \approx 0.8333$ is less than 1, which means the function will decay over time. The rate of decay is determined by the factor $\frac{t}{9}$, which indicates that the function decreases more slowly.

### Understanding the Decay:

- When $t = 0$, $y(0) = 60000 \cdot \left(\frac{5}{6}\right)^0 = 60000$.
- As $t \to \infty$, $\left(\frac{5}{6}\right)^{\frac{t}{9}} \to 0$, so $y(t) \to 0$.

### Summary:

The function $y(t) = 60000 \left(\frac{5}{6}\right)^{\frac{t}{9}}$ represents an exponential decay where the initial value is 60,000, and it decreases with a decay factor of $\frac{5}{6}$ every 9 units of time $t$.

Would you like more details on any part of this explanation, or do you have any specific questions?

### Relative Questions to Explore:

1. How does changing the base of the exponential decay affect the rate of decay?
2. What is the half-life of the given function, and how can we calculate it?
3. How would this function behave differently if the base were greater than 1?
4. What are some real-world phenomena that this type of exponential decay could model?
5. Can you derive the function for the rate of change $y'(t)$ of this decay?

**Tip:** Exponential decay functions are widely used in fields like physics, biology, and finance to model processes that decrease over time, such as radioactive decay, population decline, and depreciation of assets.

y(t) = 60000 e^{\frac{t}{9} \ln\left(\frac{5}{6}\right)}

Explore how to simplify and analyze the exponential decay function y(t) = 60000 e^(t/9 ln(5/6)). This problem highlights key concepts in exponential decay and logarithmic functions, suitable for Grades 10-12.

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