The following table shows the number of US cell phone subscribers in millions for years since 1995. Year # susbscribers 0 33.786 1 44.043 2 55.312 3 69.209 4 86.047 5 109.478 6 128.375 7 140.766 8 158.722 9 182.14 10 207.896 11 233.041 12 255.396 13 270.334 14 285.646 15 302.859 Let C(t) be the number of US cell phone subscribers in millions in t years since 1995. A linear model for the data is C(t)= 18.997t + 17.711. Use the above scatter plot to decide whether the linear model fits the data well. The function is a good model for the data. The function is not a good model for the data Estimate the number of subscribers in 2016. millions Use the model to predict the year in which the number of subscribers will be 321 million.

Let's break the problem down into two parts: determining if the linear model fits the data and using the linear model to estimate future values.

### Part 1: Evaluating if the linear model fits the data

You provided the linear model:
\[
C(t) = 18.997t + 17.711
\]
Where \( t \) is the number of years since 1995, and \( C(t) \) is the number of US cell phone subscribers in millions.

#### To assess if the model fits the data well:
We can compare the values predicted by the model to the actual values from the table.

For example, let's calculate the predicted number of subscribers for a few years using the linear model.

- For \( t = 0 \) (1995):
  \[
  C(0) = 18.997(0) + 17.711 = 17.711 \text{ million}
  \]
  (Actual: 33.786 million, so the model underestimates.)

- For \( t = 5 \) (2000):
  \[
  C(5) = 18.997(5) + 17.711 = 112.696 \text{ million}
  \]
  (Actual: 109.478 million, so it's quite close.)

- For \( t = 10 \) (2005):
  \[
  C(10) = 18.997(10) + 17.711 = 207.681 \text{ million}
  \]
  (Actual: 207.896 million, a very close estimate.)

We can conclude that for some years, the linear model fits the data well, but for others (especially early years), it might not fit perfectly. Overall, the model seems to approximate the growth reasonably well, especially in the middle years.

### Part 2: Estimating the number of subscribers in 2016

To estimate the number of subscribers in 2016, we need to find \( C(21) \), since 2016 is 21 years after 1995.

\[
C(21) = 18.997(21) + 17.711
\]

Let's compute this:

\[
C(21) = 398.937 + 17.711 = 416.648 \text{ million}
\]

So, the estimated number of subscribers in 2016 is approximately **416.648 million**.

### Part 3: Predicting the year when the number of subscribers will be 321 million

To predict the year in which the number of subscribers will reach 321 million, we set \( C(t) = 321 \) and solve for \( t \).

\[
321 = 18.997t + 17.711
\]

Now, solve for \( t \):

\[
321 - 17.711 = 18.997t
\]

\[
303.289 = 18.997t
\]

\[
t = \frac{303.289}{18.997} \approx 15.96
\]

So, \( t \approx 16 \), which means the model predicts the number of subscribers will reach 321 million around **2011** (16 years after 1995).

### Summary:
1. The model is a reasonable fit for the data, though it underestimates early years.
2. The estimated number of subscribers in 2016 is about **416.648 million**.
3. The model predicts that the number of subscribers will reach **321 million in 2011**.

Do you have any questions or would you like further details?

---

Here are five related questions:
1. How would the prediction change if the growth rate was higher than 18.997?
2. What happens if we use a quadratic model instead of a linear one?
3. How does the model behave beyond 2016? Will it continue to be accurate?
4. How would you adjust the model to better fit the earlier years?
5. How can we evaluate the goodness of fit of the linear model statistically?

**Tip**: To determine if a linear model fits well, calculate the residuals (differences between actual and predicted values) and check for any consistent patterns.

Explore the growth of US cell phone subscribers from 1995 using a linear model C(t) = 18.997t + 17.711. Learn how to estimate future subscribers, assess model accuracy, and predict when the number of subscribers will reach 321 million. Suitable for Grades 9-12.

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