Math Problem Statement
Solution
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Problem Analysis
The piecewise function describes the snowfall in inches over time () since 12 PM:
\begin{cases} 0.5t & \text{for } 0 \leq t < 2, \\ 2t - 3 & \text{for } 2 \leq t \leq 5, \\ 7 & \text{for } t > 5. \end{cases}$$ #### (a) **Graph the function $$s(t)$$:** To graph: 1. Break $$t$$ into intervals based on the function: - $$0 \leq t < 2$$: Linear with slope 0.5. - $$2 \leq t \leq 5$$: Linear with slope 2. - $$t > 5$$: Constant $$s(t) = 7$$. 2. Calculate $$s(t)$$ at key points: - For $$t = 0, 1, 2, 3, 5, 6$$, calculate: $$s(0) = 0, \, s(1) = 0.5, \, s(2) = 1, \, s(3) = 3, \, s(5) = 7, \, s(6) = 7.$$ 3. Plot these points and connect as piecewise linear sections. #### (b) **How much snow is on the ground at 3 PM?** At 3 PM ($$t = 3$$): - $$s(t) = 2t - 3$$, for $$2 \leq t \leq 5$$. - $$s(3) = 2(3) - 3 = 6 - 3 = 3$$ inches. #### (c) **When did it snow the hardest?** The rate of snowfall corresponds to the slope of $$s(t)$$: - $$0.5$$ for $$0 \leq t < 2$$, - $$2$$ for $$2 \leq t \leq 5$$, - $$0$$ for $$t > 5$$. Thus, snowfall was hardest (fastest) during $$2 \leq t \leq 5$$. #### (d) **What time did the snow stop?** The snowfall stops when $$t > 5$$, as $$s(t) = 7$$ remains constant (no accumulation). Hence, it stopped at **5 PM**. #### (e) **How much snow fell in total?** Calculate total snowfall: 1. From $$0 \leq t < 2$$: Area of a triangle with height $$1$$ (at $$t = 2$$): $$\text{Area} = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 2 \cdot 1 = 1.$$ 2. From $$2 \leq t \leq 5$$: Trapezoid with bases $$1$$ and $$7$$, and height $$3$$: $$\text{Area} = \frac{1}{2} \cdot (\text{base}_1 + \text{base}_2) \cdot \text{height} = \frac{1}{2} \cdot (1 + 7) \cdot 3 = 12.$$ 3. Total snowfall = $$1 + 12 = 13$$ inches. --- ### **Final Answers** (a) Graph: Plot the points and connect linearly as described. (b) **3 PM snowfall:** $$3$$ inches. (c) **Hardest snowfall:** $$2 \leq t \leq 5$$. (d) **Snow stopped:** 5 PM. (e) **Total snow:** $$13$$ inches. --- Would you like more details or explanations? Here are 5 related questions to expand on the topic: 1. How do we calculate areas under piecewise-linear functions? 2. Can this problem be solved using integrals? Why or why not? 3. How does the rate of snowfall relate to the slope of $$s(t)$$? 4. What changes would the graph exhibit if $$s(t)$$ included negative snowfall rates? 5. How can we use this model to predict snowfall for other scenarios? **Tip:** To graph piecewise functions, always mark the domain clearly and calculate values at key intervals!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Rate of Change
Area under a Curve
Formulas
Piecewise function definition for s(t): s(t) = { 0.5t for 0 ≤ t < 2; 2t - 3 for 2 ≤ t ≤ 5; 7 for t > 5 }
Area of a triangle = 1/2 * base * height
Area of a trapezoid = 1/2 * (base1 + base2) * height
Theorems
Linear Function Properties
Rate of Change Interpretation
Suitable Grade Level
Grades 9-12
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