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The image shows a mathematical problem related to an exponential decay model. Here's a breakdown of the information and how to solve it:

### Problem Breakdown

1. **Graph Description**: The graph plots the number of visitors \( y \) against the number of weeks since opening \( x \). It shows a declining trend, indicating fewer visitors over time.

2. **Exponential Decay Model**: The equation provided is:
   \[
   y = 75 \cdot (0.85)^x
   \]
   where:
   - \( y \) represents the number of visitors (in thousands),
   - \( x \) represents the number of weeks since opening.

3. **Questions**:
   - **a.** Use the exponential model to approximate the number of visitors after 10 weeks. Round to the nearest whole number.
   - **b.** Determine approximately when there will be 5,000 visitors.

### Solutions

1. **Approximating Visitors After 10 Weeks**:
   Plug \( x = 10 \) into the model equation:
   \[
   y = 75 \cdot (0.85)^{10}
   \]

   Calculating \( (0.85)^{10} \):
   \[
   (0.85)^{10} \approx 0.1969
   \]

   So:
   \[
   y = 75 \cdot 0.1969 \approx 14.7675
   \]
   Rounding to the nearest whole number gives approximately 15,000 visitors.

2. **Determining When There Will Be 5,000 Visitors**:
   Set \( y = 5 \) (since \( y \) is in thousands):
   \[
   5 = 75 \cdot (0.85)^x
   \]

   Solving for \( x \):
   \[
   (0.85)^x = \frac{5}{75} = \frac{1}{15}
   \]

   Taking the natural logarithm on both sides:
   \[
   \ln((0.85)^x) = \ln\left(\frac{1}{15}\right)
   \]

   \[
   x \cdot \ln(0.85) = \ln(1) - \ln(15)
   \]

   \[
   x \cdot \ln(0.85) = -\ln(15)
   \]

   \[
   x = \frac{-\ln(15)}{\ln(0.85)}
   \]

   Calculating the values:
   \[
   \ln(15) \approx 2.7081 \quad \text{and} \quad \ln(0.85) \approx -0.1625
   \]

   \[
   x \approx \frac{2.7081}{0.1625} \approx 16.67
   \]

   Rounding gives approximately 17 weeks.

### Final Answers
- **a.** The number of visitors after 10 weeks is approximately **15,000**.
- **b.** The number of visitors will reach 5,000 after approximately **17 weeks**.

Would you like more details or have any questions about the solution?

### Relative Questions to Expand Understanding
1. How does the base of the exponential function (0.85) affect the rate of decay in visitor numbers?
2. What would the model look like if the base were greater than 1? How would the graph change?
3. How can logarithms be used to solve different types of exponential equations?
4. What other real-world phenomena can be modeled using exponential decay?
5. How would changes to the initial value (75 in the model) impact the behavior of the graph?

### Tip
To solve exponential equations, taking the logarithm of both sides is a powerful technique to simplify and solve for the unknown exponent.

The graph shows the number of visitors who attended a new amusement park since opening over several weeks. Use the exponential decay model y = 75 * (0.85)^x to solve the following questions: a) Approximate the number of visitors after 10 weeks, rounding to the nearest whole number. b) Determine approximately when there will be 5,000 visitors.

Explore an exponential decay problem involving the decline of visitors to an amusement park over time. Learn how to use the decay formula y = 75 * (0.85)^x to calculate the number of visitors after a given period and determine when the visitors will reach 5,000. Suitable for high school math students.

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