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.