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We are given the piecewise function \( v(t) \), which models the average speed (in meters per second) as a function of time \( t \), where:

\[
v(t) =
\begin{cases} 
at, & 0 < t \leq 60 \\
120, & 60 < t \leq 200 \\
bt + 1740, & 200 < t \leq 360 
\end{cases}
\]

We also know:
- At \( t = 20 \) seconds, the average speed \( v(20) = 100 \, \text{m/s} \)
- At \( t = 4.5 \, \text{minutes} = 270 \, \text{seconds} \), the average speed \( v(270) = 120 \, \text{m/s} \)

### Step 1: Solve for \( a \)
We are in the first part of the function, \( v(t) = at \), for \( 0 < t \leq 60 \). Plugging in \( t = 20 \) and \( v(20) = 100 \), we get:

\[
v(20) = a \cdot 20 = 100
\]

Solving for \( a \):

\[
a = \frac{100}{20} = 5
\]

### Step 2: Solve for \( b \)
For \( t = 270 \) seconds, we use the third part of the function, \( v(t) = bt + 1740 \), which is valid for \( 200 < t \leq 360 \). We know \( v(270) = 120 \), so:

\[
v(270) = b \cdot 270 + 1740 = 120
\]

Simplifying:

\[
270b + 1740 = 120
\]

\[
270b = 120 - 1740 = -1620
\]

\[
b = \frac{-1620}{270} = -6
\]

### Final Answers:
- \( a = 5 \)
- \( b = -6 \)

Would you like further details or have any questions?

Here are five follow-up questions to deepen understanding:
1. How would the value of \( a \) change if the speed at 20 seconds were different?
2. What happens to the speed for \( t \in (60, 200] \)?
3. How would you interpret the third piece of the function, \( bt + 1740 \), in a real-world context?
4. What would be the impact on \( v(t) \) if \( b \) were positive instead of negative?
5. How could you use this function to determine the total distance traveled over the interval \( 0 < t \leq 360 \)?

**Tip:** When working with piecewise functions, always check the domain of each piece before substituting values!

The function v(t) models the average speed as a function of t, with three different conditions for time intervals. The average speed at 20 seconds is 100 m/s, and at 4.5 minutes (270 seconds) is 120 m/s. Solve for a and b.

Learn how to solve a piecewise function for a and b using given average speeds at different time intervals. In this problem, the function models speed with different conditions for time intervals, with speeds at 20 seconds and 4.5 minutes given. Detailed step-by-step solution suitable for Grades 9-11.

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