A metropolitan waterworks normally charges a flat monthly rate of $20 for up to 100

gallons of water consumption and a rate of $5 per gallon of water for consumption in 

excess of 100 gallons. 

(a) Write a piecewise definition of the monthly cost function, 𝐶(𝑥) where 𝑥 is in 

gallons.

(b) Find lim 𝑥→100

𝐶(𝑥).


### (a) **Piecewise definition of the monthly cost function, $ C(x) $**

Let $ C(x) $ represent the monthly cost as a function of the number of gallons of water consumed, $ x $. The cost structure is defined as:

- If $ x \leq 100 $ gallons, the cost is a flat $20.
- If $ x > 100 $ gallons, the cost includes the flat $20 for the first 100 gallons, plus $5 for each gallon above 100 gallons.

Thus, the piecewise function can be written as:
\[
C(x) =
\begin{cases} 
20 & \text{if } x \leq 100 \\
20 + 5(x - 100) & \text{if } x > 100
\end{cases}
\]

### (b) **Limit as $ x \to 100 $**

We want to find $ \lim_{x \to 100} C(x) $.

- When $ x = 100 $, we are within the first part of the piecewise function, so $ C(100) = 20 $.

- As $ x \to 100^+ $ (from the right), the cost will still approach $ 20 $ because when $ x > 100 $, the extra charge $ 5(x - 100) $ becomes very small as $ x $ approaches 100 from above.

Thus:
\[
\lim_{x \to 100} C(x) = 20
\]

This indicates that the function is continuous at $ x = 100 $, and the cost is $20 for 100 gallons, whether we approach from below or above.

---

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How does the function $ C(x) $ behave as $ x \to \infty $?
2. What is the slope of $ C(x) $ for $ x > 100 $?
3. Is the function $ C(x) $ differentiable at $ x = 100 $?
4. How would the cost function change if the flat rate was for 150 gallons instead?
5. Can you find the cost if the water consumption is 150 gallons?

**Tip**: When working with piecewise functions, always check for continuity and differentiability at the boundary points (e.g., $ x = 100 $ here).

A metropolitan waterworks normally charges a flat monthly rate of $20 for up to 100 gallons of water consumption and a rate of $5 per gallon of water for consumption in excess of 100 gallons.

(a) Write a piecewise definition of the monthly cost function, C(x), where x is in gallons.
(b) Find lim x→100 C(x).

Explore how to define the monthly cost function for water consumption with a piecewise formula, including calculating the limit as x approaches 100 gallons. Learn about limits, continuity, and real-world applications of piecewise functions. Suitable for students in Grades 10-12.

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