Math Problem Statement
A taxi company charges $3.00 for the first mile (or part of a mile) and 60 cents for each succeeding tenth of a mile (or part). Express the cost C(x) of a ride as a piecewise defined function of the distance x traveled (in miles) for 0 < x ≤ 2.
Solution
The problem asks for the cost of a taxi ride as a piecewise function, given the charges:
- $3.00 for the first mile (or part of a mile)
- 60 cents for each succeeding tenth of a mile (or part) thereafter.
We are asked to define as a piecewise function of the distance traveled , for .
Let's break down the problem:
- For , the cost is $3.00 for any distance up to 1 mile.
- For distances beyond 1 mile, every additional tenth of a mile adds 60 cents.
We can express the piecewise function as follows:
\begin{cases} 3.00 & \text{if } 0 < x \leq 1, \\ 3.00 + 0.60 & \text{if } 1 < x \leq 1.1, \\ 3.00 + 2(0.60) & \text{if } 1.1 < x \leq 1.2, \\ \vdots & \\ 3.00 + 9(0.60) & \text{if } 1.9 < x \leq 2. \end{cases}$$ Simplifying each expression: $$C(x) = \begin{cases} 3.00 & \text{if } 0 < x \leq 1, \\ 3.60 & \text{if } 1 < x \leq 1.1, \\ 4.20 & \text{if } 1.1 < x \leq 1.2, \\ 4.80 & \text{if } 1.2 < x \leq 1.3, \\ 5.40 & \text{if } 1.3 < x \leq 1.4, \\ 6.00 & \text{if } 1.4 < x \leq 1.5, \\ 6.60 & \text{if } 1.5 < x \leq 1.6, \\ 7.20 & \text{if } 1.6 < x \leq 1.7, \\ 7.80 & \text{if } 1.7 < x \leq 1.8, \\ 8.40 & \text{if } 1.8 < x \leq 1.9, \\ 9.00 & \text{if } 1.9 < x \leq 2. \end{cases}$$ This is the piecewise function representing the cost $$C(x)$$ in dollars for $$0 < x \leq 2$$ miles. Would you like further details or explanations? Here are some related questions for practice: 1. What is the cost for a distance of 1.5 miles? 2. How would the function change if the cost for each tenth of a mile increased to 80 cents? 3. How would the function change if the base fare was reduced to $2.50? 4. What is the cost for exactly 2 miles? 5. Can this piecewise function be written using a floor function instead of separate cases? **Tip:** Piecewise functions are useful in real-world scenarios where different rules apply to different ranges of a variable.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Step Functions
Formulas
C(x) = 3.00 for 0 < x ≤ 1, C(x) = 3.00 + 0.60n for x > 1, where n is the number of additional 0.1 mile increments
Theorems
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Suitable Grade Level
Grades 9-12