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Let's analyze the problem and calculate the average rates of change for the intervals specified.

The function given is:
\[
d(t) = -0.086t^3 + 3.71t^2 - 53.7t + 643
\]
where \( t \) is the number of years after 1995. Therefore:
- \( t = 1 \) corresponds to 1996.
- \( t = 11 \) corresponds to 2006.
- \( t = 23 \) corresponds to 2018.

### Part (b): Calculating Average Rates of Change
The average rate of change over an interval \([a, b]\) for a function \(f(t)\) is calculated as:
\[
\frac{f(b) - f(a)}{b - a}
\]

#### Step 1: Evaluate \( d(t) \) at the Required Points
1. **For \( t = 1 \) (1996):** \( d(1) \)
2. **For \( t = 11 \) (2006):** \( d(11) \)
3. **For \( t = 23 \) (2018):** \( d(23) \)

Let's calculate each of these values and then compute the average rate of change for each interval.

#### Interval 1: 1996 to 2006
\[
\text{Average rate of change} = \frac{d(11) - d(1)}{11 - 1}
\]

#### Interval 2: 2006 to 2018
\[
\text{Average rate of change} = \frac{d(23) - d(11)}{23 - 11}
\]

I'll calculate these values to provide exact answers.Here are the results:

1. \( d(1) \approx 593 \)
2. \( d(11) \approx 387 \)
3. \( d(23) \approx 324 \)

### Average Rates of Change:
- From 1996 to 2006: approximately **-21 theaters per year**.
- From 2006 to 2018: approximately **-5 theaters per year**.

These values show a decrease in the rate of decline over time, indicating that the number of drive-in theaters is dropping less sharply in recent years.

Would you like a further explanation on these calculations or more details?

### Related Questions
1. How do you interpret the behavior of the function \( d(t) \) as \( t \) increases?
2. What could be some factors contributing to the decline in drive-in theaters?
3. How would the results change if we considered other intervals, say from 1996 to 2018?
4. Could a similar model be applied to other types of theaters or entertainment venues?
5. What does the rate of change tell us about the trend in drive-in theaters over time?

### Tip
The average rate of change helps summarize how a function behaves over a specific interval, which is useful for interpreting trends.

The number of drive-in movie theaters in the United States from 1995 to 2018 can be modeled by the function d(t) = -0.086t^3 + 3.71t^2 - 53.7t + 643, where t is the number of years after 1995. Calculate the average rates of change in the number of drive-in movie theaters from 1996 to 2006 and from 2006 to 2018. Describe the behavior of the graph on this interval.

Analyze the decline in the number of drive-in theaters in the United States from 1995 to 2018 using a polynomial model. Calculate the average rate of change from 1996 to 2006 and from 2006 to 2018, with a focus on interpreting trends in drive-in theater popularity.

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