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Let's break down and solve the problem step by step.

### a) Construct the piecewise function

The problem gives us a tiered rate structure for electricity usage. We need to express the cost $ C(x) $ as a piecewise function based on the total electricity consumption $ x $ in kilowatt-hours (kWh):

1. **For the first 200 kWh:**
   - Cost per kWh: 22 cents
   - Total cost: $ 0.22x $ for $ 0 \leq x \leq 200 $

2. **For the next 100 kWh (i.e., from 201 kWh to 300 kWh):**
   - Cost per kWh: 33 cents
   - Total cost for these 100 kWh: $ 0.33(x - 200) + 0.22(200) = 0.33(x - 200) + 44 $ for $ 200 < x \leq 300 $

3. **For the next 300 kWh (i.e., from 301 kWh to 600 kWh):**
   - Cost per kWh: 52 cents
   - Total cost for these 300 kWh: $ 0.52(x - 300) + 0.33(100) + 44 = 0.52(x - 300) + 77 $ for $ 300 < x \leq 600 $

4. **For any usage above 600 kWh:**
   - Cost per kWh: 55 cents
   - Total cost for the remaining kWh: $ 0.55(x - 600) + 0.52(300) + 77 = 0.55(x - 600) + 233 $ for $ x > 600 $

Thus, the piecewise function is:

\[
C(x) = 
\begin{cases} 
0.22x & 0 \leq x \leq 200 \\
0.33(x - 200) + 44 & 200 < x \leq 300 \\
0.52(x - 300) + 77 & 300 < x \leq 600 \\
0.55(x - 600) + 233 & x > 600 
\end{cases}
\]

### b) Graph the piecewise functions

For the graph, each section will be linear within the specified interval, with the following slopes and y-intercepts:
- $ 0.22x $ from $ 0 \leq x \leq 200 $
- $ 0.33(x - 200) + 44 $ from $ 200 < x \leq 300 $
- $ 0.52(x - 300) + 77 $ from $ 300 < x \leq 600 $
- $ 0.55(x - 600) + 233 $ from $ x > 600 $

The graph will have "kinks" at $ x = 200 $, $ x = 300 $, and $ x = 600 $ where the slope changes.

### c) Calculate the charge for 724 kWh

Amran used 724 kWh, which is above the 600 kWh threshold, so we use the last piece of the piecewise function:

\[
C(724) = 0.55(724 - 600) + 233
\]

Calculating:

\[
C(724) = 0.55 \times 124 + 233 = 68.2 + 233 = 301.2 \text{ cents}
\]

So, the total charge is **301.2 cents**, or **$3.012**.

### Summary
- The piecewise function has been established, with breakpoints at 200, 300, and 600 kWh.
- The function was graphed, showing linear segments with different slopes.
- The cost for 724 kWh is calculated as 301.2 cents or $3.012.

Would you like to see more details or have any further questions?

#### Relative Questions:
1. How do the breakpoints in the piecewise function affect the graph?
2. What would the charge be if Amran used exactly 600 kWh?
3. How does the slope change after each breakpoint?
4. What is the significance of piecewise functions in real-life scenarios like electricity billing?
5. Can the piecewise function be integrated to find the total cost for a given range of kWh usage?

**Tip:** Always double-check your piecewise functions by calculating a few points manually to ensure they match the expected cost structure.

For a monthly electric tariff, Tenaga Nasional charged 22 cents/kWh for the first 200 kWh usage, 33 cents/kWh for the next 100 kWh, 52 cents/kWh for 300 kWh afterwards, and another 55 cents/kWh for the remainder electric usage. a) Construct the piecewise-functions for the problem. b) Graph the piecewise-functions. c) Find the electric charge that Amran must pay if he used 724 kWh in one month.

Learn how to construct and graph piecewise functions based on an electricity tariff calculation. This problem covers key algebraic concepts, including piecewise functions and linear equations. The solution includes a step-by-step breakdown, suitable for students in Grades 10-12.

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