Math Problem Statement

A tricycle ride cost 10 for the first 2 kilometers and each additional kilometers adds 8 to the fare. Use piecewise function to the represent the tricycle fare in terms of the distance d in kilometers

Solution

To represent the tricycle fare as a piecewise function in terms of the distance dd in kilometers, we need to account for the fare structure:

  1. The first 2 kilometers cost 10.
  2. Each additional kilometer beyond 2 kilometers costs an additional 8 per kilometer.

Let's define the fare function f(d)f(d):

10 & \text{if } 0 \leq d \leq 2 \\ 10 + 8(d - 2) & \text{if } d > 2 \end{cases}$$ This function can be broken down as follows: - For $$d$$ between 0 and 2 kilometers, the fare is constant at 10. - For $$d$$ greater than 2 kilometers, the fare starts at 10 for the first 2 kilometers, plus an additional 8 for each kilometer beyond the first 2 kilometers. Let's simplify the second part of the piecewise function: $$f(d) = 10 + 8(d - 2)$$ $$f(d) = 10 + 8d - 16$$ $$f(d) = 8d - 6$$ Thus, the piecewise function can be written as: $$f(d) = \begin{cases} 10 & \text{if } 0 \leq d \leq 2 \\ 8d - 6 & \text{if } d > 2 \end{cases}$$ This piecewise function accurately represents the tricycle fare in terms of the distance traveled $$d$$. Would you like more details or have any questions about this piecewise function? Here are some related questions to expand your understanding: 1. How would the piecewise function change if the initial fare was for 3 kilometers instead of 2? 2. What is the fare for a 5-kilometer ride? 3. How do you graph a piecewise function like this? 4. How can you use this piecewise function to find the fare for a non-integer distance, such as 3.5 kilometers? 5. What is the fare for a 2-kilometer ride according to this function? 6. How does the function change if the cost per additional kilometer was 10 instead of 8? 7. What are the domain and range of this piecewise function? 8. How do you write the piecewise function if the base fare changes to 12 for the first 2 kilometers and 9 for each additional kilometer? **Tip:** When dealing with piecewise functions, always carefully define the intervals and ensure each piece of the function correctly matches its corresponding interval.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions

Formulas

Piecewise function definition

Theorems

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Suitable Grade Level

High School

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