Erik drives 30 mph for 30 minutes, then he accelrates at a constant rate for another 0 minutes until he reaches a rate of 60 mph. Write a piecewise function to model this situation

To model Erik's driving situation, we can create a piecewise function based on two phases:

1. **Constant speed** for the first 30 minutes at 30 mph.
2. **Acceleration** for the next 10 minutes until he reaches 60 mph.

Let the function \( f(t) \) represent Erik's speed at time \( t \), where \( t \) is in minutes, and we want to express it as a piecewise function. Here's the breakdown:

- **First phase (0 to 30 minutes)**: Erik drives at a constant speed of 30 mph for 30 minutes.
  
  For \( 0 \leq t \leq 30 \), the speed is constant at 30 mph:
  \[
  f(t) = 30
  \]

- **Second phase (30 to 40 minutes)**: Erik accelerates from 30 mph to 60 mph over 10 minutes. The acceleration is constant, so this is linear acceleration.

  The equation for a line can be written as:
  \[
  f(t) = mt + b
  \]
  where \( m \) is the slope (rate of change of speed) and \( b \) is the y-intercept (initial speed at \( t = 30 \)).

  To find \( m \), we use:
  \[
  m = \frac{\Delta \text{speed}}{\Delta \text{time}} = \frac{60 - 30}{40 - 30} = \frac{30}{10} = 3
  \]
  So, the slope is 3 mph per minute. Now we find the equation of the line during this phase, starting at \( t = 30 \) with \( f(30) = 30 \):
  \[
  f(t) = 3(t - 30) + 30 = 3t - 90 + 30 = 3t - 60
  \]

Thus, the piecewise function can be written as:
\[
f(t) =
\begin{cases}
30 & \text{for } 0 \leq t \leq 30 \\
3t - 60 & \text{for } 30 < t \leq 40
\end{cases}
\]

### Summary:
The piecewise function modeling Erik's speed is:
\[
f(t) =
\begin{cases}
30 & \text{if } 0 \leq t \leq 30 \\
3t - 60 & \text{if } 30 < t \leq 40
\end{cases}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:
1. How would you graph this piecewise function?
2. What is Erik’s acceleration in meters per second squared during the second phase?
3. How far does Erik travel during each phase?
4. How would you modify this function if Erik reached 60 mph in 15 minutes instead?
5. What would the speed function look like if the acceleration was non-constant?

**Tip**: Piecewise functions can be used to describe real-world situations with multiple phases, like changes in speed or direction!

Erik drives 30 mph for 30 minutes, then he accelerates at a constant rate for another 10 minutes until he reaches a rate of 60 mph. Write a piecewise function to model this situation.

Explore how to write a piecewise function to model Erik's driving scenario, where he travels at 30 mph for 30 minutes and then accelerates to 60 mph over 10 minutes. This example covers key concepts like piecewise functions, linear acceleration, and slope calculation. Suitable for Grades 9-12 students learning advanced algebra.

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