Math Problem Statement

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 afterward, and another 55 cents/kWh for the remainder electric usage. Find the electric charge for 724 kWh usage.

Solution

Let's break down the problem into parts and address each question:

Given:

  • First 200 kWh → 22 cents/kWh
  • Next 100 kWh → 33 cents/kWh
  • Next 300 kWh → 52 cents/kWh
  • Any usage beyond 600 kWh → 55 cents/kWh

Part (a): Construct the piecewise function

Let C(x)C(x) represent the total cost of using xx kWh in one month. The piecewise function can be defined based on the given rates:

\begin{cases} 22x & \text{if } 0 \leq x \leq 200, \\ 22(200) + 33(x - 200) & \text{if } 200 < x \leq 300, \\ 22(200) + 33(100) + 52(x - 300) & \text{if } 300 < x \leq 600, \\ 22(200) + 33(100) + 52(300) + 55(x - 600) & \text{if } x > 600. \end{cases}$$ ### Part (b): Graph the piecewise function To graph this, you can plot different segments of the function as straight lines: - The first segment will have a slope of 22 up to $$x = 200$$. - The second segment will have a steeper slope of 33 for $$200 < x \leq 300$$. - The third segment will have a slope of 52 for $$300 < x \leq 600$$. - Finally, the last segment will have the highest slope of 55 for $$x > 600$$. ### Part (c): Find the electric charge for 724 kWh usage To find the total charge for 724 kWh, we follow the piecewise function: 1. For the first 200 kWh: $$200 \times 22 = 4400 \text{ cents} = 44 \text{ dollars}$$ 2. For the next 100 kWh (from 200 to 300 kWh): $$100 \times 33 = 3300 \text{ cents} = 33 \text{ dollars}$$ 3. For the next 300 kWh (from 300 to 600 kWh): $$300 \times 52 = 15600 \text{ cents} = 156 \text{ dollars}$$ 4. For the remaining 124 kWh (beyond 600 kWh): $$124 \times 55 = 6820 \text{ cents} = 68.20 \text{ dollars}$$ Now, sum up the charges: $$44 + 33 + 156 + 68.20 = 301.20 \text{ dollars}$$ Thus, Amran must pay **$301.20** for using 724 kWh in one month. --- Would you like further details or have any questions? ### Additional Related Questions: 1. How does the slope of each segment in the piecewise function relate to the cost increase? 2. What happens if the consumption exceeds 1000 kWh? How would the cost increase? 3. How could you write a generalized function for this type of tariff structure? 4. How do we handle the domain of each segment in a piecewise function? 5. How would the charges change if the rates per kWh were adjusted for inflation? ### Tip: When solving piecewise functions, always carefully determine which range your input falls into and apply the corresponding rule.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Equations
Tariff Calculation

Formulas

C(x) = 22x if 0 ≤ x ≤ 200
C(x) = 22(200) + 33(x - 200) if 200 < x ≤ 300
C(x) = 22(200) + 33(100) + 52(x - 300) if 300 < x ≤ 600
C(x) = 22(200) + 33(100) + 52(300) + 55(x - 600) if x > 600

Theorems

Piecewise Function Definition
Linear Cost Model

Suitable Grade Level

Grades 10-12

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