We are given an exponential model $s = 0.841 \times (1.037)^t$, where $s$ represents the cost of a stamp in cents and $t$ represents the time in years since 1900.

Part b: Calculate the model cost for a stamp in 1988

In 1988, $t$ is the number of years since 1900: $t = 1988 - 1900 = 88$

Substitute $t = 88$ into the exponential model $s = 0.841 \times (1.037)^t$: $s = 0.841 \times (1.037)^{88}$

To calculate:

$s = 0.841 \times (1.037)^{88} \approx 0.841 \times 14.241 \approx 11.98$

So, the model gives a cost of approximately 12 cents for a stamp in 1988 (when rounded to the nearest cent). The actual cost in 1988 was 25 cents.

Part c: Plot the data and the exponential model

To plot the data and the exponential model, we can:

1. Plot the given data points $(t, s)$ from the table:

$t = \{19, 32, 58, 71, 78, 85, 95, 102, 109, 116\}$ $s = \{2, 3, 4, 8, 15, 22, 32, 37, 44, 47\}$

2. Overlay the exponential model $s = 0.841 \times (1.037)^t$ for a range of $t$ values (e.g., 0 to 116).

Would you like me to plot the graph for you and provide more details?

Here are 5 related questions for further exploration:

1. How would changing the base of the exponential function affect the predictions for future years?
2. What would the model predict for the cost of a stamp in 2024?
3. How does the difference between the model's prediction and actual data in 1988 suggest a need for a different model?
4. How would a linear model compare to this exponential model for predicting stamp prices?
5. What would the stamp price have been in 1950 according to the model?

Tip: Exponential models are particularly useful for representing growth that accelerates over time, such as inflation-adjusted costs.