In 2008​, people in a particular region spent an average of 18 minutes daily on their mobile devices. By 2015 that figure was up to 183 minutes daily. Assuming the increase in usage was​ constant, the number of minutes used can be calculated by the equation y=23.6x+18​, where x represents the number of years after 2008 and y represents the number of daily minutes spent on mobile devices. Complete parts​ (a) through​ (e) below.

Let's break down the given equation and answer each part. The equation provided is:

\[
y = 23.6x + 18
\]

where:

- \( y \) = the number of daily minutes spent on mobile devices.
- \( x \) = the number of years after 2008.

### (a) Verify that in 2015, the average number of minutes spent daily on mobile devices was 183.

To check this, we need to calculate \( y \) for \( x = 2015 - 2008 = 7 \) years after 2008:

\[
y = 23.6(7) + 18
\]
\[
y = 165.2 + 18 = 183.2
\]

This matches the given figure of 183 minutes, so the equation is accurate.

### (b) Find the number of daily minutes people spent on mobile devices in 2010.

For 2010, \( x = 2010 - 2008 = 2 \):

\[
y = 23.6(2) + 18
\]
\[
y = 47.2 + 18 = 65.2
\]

So, in 2010, people spent approximately **65.2 minutes** daily on mobile devices.

### (c) Find the year when people spent an average of 150 minutes daily on their mobile devices.

We need to solve for \( x \) when \( y = 150 \):

\[
150 = 23.6x + 18
\]
Subtract 18 from both sides:

\[
132 = 23.6x
\]

Solve for \( x \):

\[
x = \frac{132}{23.6} \approx 5.59
\]

Since \( x \) represents the number of years after 2008, we add 5.59 to 2008:

\[
2008 + 5.59 \approx 2013.59
\]

So, people spent an average of 150 minutes daily sometime in late **2013**.

### (d) Interpret the slope of the equation in the context of the problem.

The slope of the equation is **23.6**. This represents the average increase in the number of daily minutes spent on mobile devices each year. In other words, people spent an additional **23.6 minutes per year** on their mobile devices, on average.

### (e) Interpret the y-intercept of the equation in the context of the problem.

The y-intercept of the equation is **18**, which corresponds to the value of \( y \) when \( x = 0 \). This means that in 2008 (the base year), people spent an average of **18 minutes daily** on their mobile devices.

---

Would you like further details or have any questions about these results?

Here are 5 related questions for you:
1. How would the equation change if the increase in daily mobile usage was faster than 23.6 minutes per year?
2. What would the usage be in 2020 based on this model?
3. Can you predict when the daily usage will surpass 250 minutes?
4. How would the interpretation change if the y-intercept was negative?
5. How can the accuracy of this linear model be tested against real-world data?

**Tip**: When interpreting slopes in real-world contexts, always remember that they represent **rate of change** over time or another variable.

In 2008, people in a particular region spent an average of 18 minutes daily on their mobile devices. By 2015 that figure was up to 183 minutes daily. Assuming the increase in usage was constant, the number of minutes used can be calculated by the equation y = 23.6x + 18, where x represents the number of years after 2008 and y represents the number of daily minutes spent on mobile devices. Complete parts (a) through (e) below.

Explore how mobile usage increased from 18 minutes in 2008 to 183 minutes in 2015 using the linear equation y = 23.6x + 18. This problem breaks down how to calculate mobile usage in different years and interpret the slope and intercept in the context of technology usage growth. Suitable for students in Grades 9-10.

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