A taxi company charges two dollars for the first mile (or part of a mile) and 20 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a function of the distance d traveled (in miles).

The cost $ C $ of a taxi ride as a function of the distance $ d $ traveled (in miles) can be expressed piecewise because the cost is structured differently for the first mile and for each subsequent part of a mile.

### Key points:
1. The first mile (or part of a mile) costs $2.
2. For any distance beyond the first mile, each additional tenth of a mile (or part) costs 20 cents.

### The function:

- For $ 0 < d \leq 1 $ (i.e., the first mile or part of it), the cost is a flat $2.
- For $ d > 1 $, the cost is $2 for the first mile, plus an additional 20 cents for each tenth of a mile (or part) after the first mile. The number of such tenths is $ 10 \times (d - 1) $.

Thus, the cost function $ C(d) $ can be expressed as:

\[
C(d) =
\begin{cases}
2 & \text{if } 0 < d \leq 1, \\
2 + 0.2 \times \lceil 10 \times (d - 1) \rceil & \text{if } d > 1,
\end{cases}
\]

where $ \lceil x \rceil $ denotes the ceiling function, which rounds $ x $ up to the nearest integer (since any part of a tenth of a mile also incurs a charge).

### Explanation:

- For distances up to 1 mile, the cost is a flat $2.
- For distances greater than 1 mile, we first subtract the initial mile ($ d - 1 $) and multiply by 10 to get the number of tenths of a mile. The ceiling function is used to ensure that any part of a tenth mile is charged as a full tenth.

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Do you want to go over how the ceiling function works in this context, or have any other questions?

Here are 5 related questions:

1. What would be the cost for a 3.2-mile trip?
2. How does the ceiling function affect rounding in this pricing model?
3. Can you derive the cost for a trip of exactly 1.7 miles?
4. What is the cost for a trip of 5 miles?
5. How would you write a Python function to calculate the cost based on distance?

**Tip**: In functions like these, ceiling and floor functions are often used to handle partial quantities that still incur full charges.

Explore how to express the cost of a taxi ride as a piecewise function of the distance traveled. The fare includes $2 for the first mile and 20 cents for each additional tenth of a mile, with detailed steps on how the ceiling function plays a role in rounding the fare. Ideal for high school students studying piecewise functions and linear equations.

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