Postal Rates The table below shows the cost s, in

cents, of a domestic first-class postage stamp in the

United States t years after 1900.
t= time, in years since 1900 19 32 58 71 78 85 95 102 109 116
s=cost of stamp 2 3 4 8 15 22 32 37 44 47
When the exponential expression is s=0.841 times (1.037)^t

b. What cost does your model give for a 1988

stamp? Report your answer to the nearest cent.

(The actual cost was 25 cents.)

c. Plot the data and the exponential model.
Use the N=Pa^t formula for part b

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.

The table below shows the cost s, in cents, of a domestic first-class postage stamp in the United States t years after 1900. t= time, in years since 1900: 19, 32, 58, 71, 78, 85, 95, 102, 109, 116; s=cost of stamp: 2, 3, 4, 8, 15, 22, 32, 37, 44, 47. When the exponential expression is s=0.841 times (1.037)^t, what cost does your model give for a 1988 stamp? Report your answer to the nearest cent. The actual cost was 25 cents. Plot the data and the exponential model.

Explore how an exponential model is used to predict the cost of a US first-class postage stamp from 1900 to 1988. The model uses s = 0.841 * (1.037)^t, and we calculate the stamp cost in 1988, compare it with the actual price of 25 cents, and plot the data. Suitable for Grades 9-12.

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