Question: User: Jamilla was moving to a new city. She researched the rates charged by the local utility company for water. She found the listing of charges below. She expects that her family may use up to 1,000 cubic feet of water each month.
• 12.70 monthly service fee
• First 300 cubic feet of water used: 3.90 per 100 cubic feet, or fraction thereof
• After the first 300 cubic feet: 5.20 per 100 cubic feet, or fraction thereof
a. Sketch a graph of the cost of Jamilla's possible water usage in one month. Be sure to consider what the cost would be for partial units such as 220 or 675 cubic feet of water.
Hint (a):
This will be a step graph. What is the cost if no water is used, but there is service?
More Help (a):
When sketching this graph, note that the price jumps from 3.90 to 5.20 after 300 cubic feet of water is used.
b. Is this graph a function? Why or why not?
Hint (b):
For each x-value, is there exactly one y-value?
c. What are the domain and range of this graph?
Hint (c):
Domain refers to the possible x-values, and range refers to the possible y-values.
1. **Problem Statement:**
Jamilla wants to know the cost of water usage up to 1000 cubic feet per month based on the given rate structure:
- Monthly service fee: $12.70$
- First 300 cubic feet: $3.90$ per 100 cubic feet or fraction thereof
- After 300 cubic feet: $5.20$ per 100 cubic feet or fraction thereof
2. **Cost Formula:**
Let $x$ be the water usage in cubic feet.
- If $0 \leq x \leq 300$, cost is:
$$\text{Cost} = 12.70 + 3.90 \times \left\lceil \frac{x}{100} \right\rceil$$
- If $x > 300$, cost is:
$$\text{Cost} = 12.70 + 3.90 \times 3 + 5.20 \times \left\lceil \frac{x - 300}{100} \right\rceil$$
Note: $\lceil y \rceil$ means rounding $y$ up to the nearest integer (fraction thereof counts as full unit).
3. **Step Graph Explanation:**
- The cost jumps at multiples of 100 cubic feet.
- For example, at $x=0$, cost is $12.70$ (service fee only).
- At $x=220$, since $220/100=2.2$, $\lceil 2.2 \rceil=3$, so cost for first 300 cubic feet is:
$$12.70 + 3.90 \times 3 = 12.70 + 11.70 = 24.40$$
- At $x=300$, cost is:
$$12.70 + 3.90 \times 3 = 24.40$$
- At $x=675$, usage beyond 300 is $675 - 300 = 375$ cubic feet.
Number of 100 cubic feet units beyond 300 is $\lceil 375/100 \rceil = \lceil 3.75 \rceil = 4$.
So cost is:
$$12.70 + 3.90 \times 3 + 5.20 \times 4 = 12.70 + 11.70 + 20.80 = 45.20$$
4. **Is this graph a function?**
- Yes, because for each water usage $x$, there is exactly one cost value.
- The cost is uniquely determined by the usage.
5. **Domain and Range:**
- Domain: $[0, 1000]$ cubic feet (usage from zero to 1000 cubic feet).
- Range: All possible costs from $12.70$ (no water used) up to the cost at 1000 cubic feet.
Calculate cost at 1000 cubic feet:
- Usage beyond 300: $1000 - 300 = 700$ cubic feet
- Units beyond 300: $\lceil 700/100 \rceil = 7$
- Total cost:
$$12.70 + 3.90 \times 3 + 5.20 \times 7 = 12.70 + 11.70 + 36.40 = 60.80$$
So range is $[12.70, 60.80]$ dollars.
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**Final answers:**
- Cost function is a step function with jumps at multiples of 100 cubic feet.
- It is a function because each input has one output.
- Domain: $[0, 1000]$
- Range: $[12.70, 60.80]$