Math Problem Statement
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.
Solution
We are given the piecewise function , which models the average speed (in meters per second) as a function of time , where:
\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!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Algebra
Linear equations
Formulas
v(t) = a * t for 0 < t <= 60
v(t) = 120 for 60 < t <= 200
v(t) = b * t + 1740 for 200 < t <= 360
Theorems
Solving linear equations
Suitable Grade Level
Grades 9-11
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