To answer your questions, let's analyze the road trip step by step:
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Determine A, B, and C in the piece-wise function: The piece-wise function given is: \[ D(t) = \begin{cases} At & \text{if } 0 \leq t \leq 1 \ B(t - 1) & \text{if } t > 1 \end{cases} \]
Given this, let’s interpret the time intervals and speeds:
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For the first hour (0 ≤ t ≤ 1), the speed was 55 mph. So for this interval: \[ D(t) = 55t \quad \text{(because the distance covered is speed × time)} \] Thus, \(A = 55\).
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For t > 1 (i.e., after the first hour), the speed changed to 70 mph. The distance covered is based on the time elapsed since the first hour. Therefore, for t > 1:
- At t = 1, the distance covered is: \[ D(1) = 55 \times 1 = 55 \text{ miles} \]
- For the second part of the function (starting from t = 1), the distance can be formulated as:
\[
D(t) = 55 + 70(t - 1)
\]
which simplifies to:
\[
D(t) = 70(t - 1) + 55 = 70t - 70 + 55 = 70t - 15
\]
Here, we identify the part \(B\) as 70 and the constant term \(C\) must be adjusted to fit the piece of the function as:
- \(C\) is the offset added from the first hour of travel, to remain consistent in units. In our case, it would be the 55 miles traveled in the first hour. The constant offset is -15 miles added, so \(C = 1\).
Therefore, we have:
- \(A = 55\)
- \(B = 70\)
- \(C = 1\)
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What does the slope of the graph at any given time represent? The slope of a distance-time graph represents speed (the rate of distance traveled over time). Thus the correct answer is "Speed."
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Calculate the distance traveled after 2.5 hours: For the first hour, they traveled 55 miles. For the next 1.5 hours (from t = 1 to t = 2.5):
- Use the formula defined for t > 1: \[ D(t) = 70(t - 1) + 55 \] For \(t = 2.5\), \[ D(2.5) = 70(2.5 - 1) + 55 \] Simplifying: \[ D(2.5) = 70(1.5) + 55 = 105 + 55 = 160 \text{ miles} \]
Therefore, the answers are:
- \(A = 55\)
- \(B = 70\)
- \(C = 1\)
Distance traveled after 2.5 hours: 160 miles.