Piecewise Cost Function for Water Usage with Limits

Math Problem Statement

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).

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:

\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).

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Piecewise Functions
Limits
Continuity

Piecewise function definition: C(x) = 20 if x ≤ 100, C(x) = 20 + 5(x - 100) if x > 100
Limit definition: lim x→100 C(x)

Limit Theorem: lim x→a f(x) = L
Continuity Theorem: A function f(x) is continuous at a point if lim x→a f(x) = f(a)

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