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The problem asks for the cost $ C(x) $ 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 $ C(x) $ as a piecewise function of the distance traveled $ x $, for $ 0 < x \leq 2 $.

Let's break down the problem:

1. For $ 0 < x \leq 1 $, the cost is $3.00 for any distance up to 1 mile.
2. For distances beyond 1 mile, every additional tenth of a mile adds 60 cents.

We can express the piecewise function as follows:

\[
C(x) = 
\begin{cases} 
3.00 & \text{if } 0 < x \leq 1, \\
3.00 + 0.60 & \text{if } 1 < x \leq 1.1, \\
3.00 + 2(0.60) & \text{if } 1.1 < x \leq 1.2, \\
\vdots & \\
3.00 + 9(0.60) & \text{if } 1.9 < x \leq 2.
\end{cases}
\]

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.

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.

Learn how to express the cost C(x) of a taxi ride as a piecewise function, where the base fare is $3.00 for the first mile and each additional tenth of a mile costs 60 cents. The function is defined for distances between 0 and 2 miles.

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