Problem 81 - Algebra and Trigonometry

Question

Westside Energy charges its electric customers a base rate of \(\$ 6.00\) per month, plus 10 \(\mathrm{cr}\) kilowatt-hour (kWh) for the first 300 \(\mathrm{kWh}\) used and 6 \(\mathrm{c}\) per kWh for all usage over 300 \(\mathrm{kWh}\) . Suppose a customer uses \(x \mathrm{kWh}\) of electricity in one month. (a) Express the monthly cost \(E\) as a piecewise-defined function of \(x .\) (b) Graph the function \(E\) for \(0 \leq x \leq 600\)

Step-by-Step Solution

Answer

E(x) is a piecewise function: \( E(x) = 6 + 0.10x \) for \( 0 \leq x \leq 300 \), and \( E(x) = 18 + 0.06x \) for \( x > 300 \).

Step 1: Define Base Usage Cost

Westside Energy has a base rate of $6.00 per month. Therefore, regardless of the electricity consumed, the customer will always pay this base rate. This is a fixed cost in the billing structure.

Step 2: Determine Cost for First 300 kWh

The cost for up to 300 kWh is \(0.10 (10 cents) per kWh. Thus, if a customer uses 300 or fewer kWh, the cost of electricity usage is \)0.10 times the number of kWh used. For the first 300 kWh, the cost can be calculated as \( \text{Cost} = 0.10x \), where \( x \leq 300 \).

Step 3: Determine Cost for Usage Over 300 kWh

If the customer uses more than 300 kWh, the cost for the first 300 kWh remains the same at \(30 (since \( 300 \times 0.10 = 30 \)). For any additional kWh over 300, the cost is \)0.06 (6 cents) per kWh. Therefore, for usage over 300 kWh, the additional cost is calculated by \( 0.06(x - 300) \).

Step 4: Combine Costs into a Piecewise Function

We can now express the total cost \( E \) as a piecewise function combining the above calculations:\[E(x) = \begin{cases} 6 + 0.10x & \text{if } 0 \leq x \leq 300 \6 + 30 + 0.06(x - 300) & \text{if } x > 300 \end{cases}\]

Step 5: Simplify the Piecewise Function

Simplifying the second case for usage over 300 kWh, substitute \( 6 + 30 = 36 \):\[E(x) = 36 + 0.06(x - 300) = 36 + 0.06x - 18 = 18 + 0.06x \] Thus, the piecewise function is:\[E(x) = \begin{cases} 6 + 0.10x & \text{if } 0 \leq x \leq 300 \18 + 0.06x & \text{if } x > 300 \end{cases}\]

Step 6: Graph the Piecewise Function

To graph the function \( E(x) \), plot the two parts:- From \( x = 0 \) to \( x = 300 \), plot \( E(x) = 6 + 0.10x \). This is a straight line starting at 6 with a slope of 0.10.- From \( x = 300 \) onwards, plot \( E(x) = 18 + 0.06x \). This starts from the point (300, 48) and has a less steep slope of 0.06. Extend the line to \( x = 600 \).

Key Concepts

Understanding Cost Structure

In the context of utility billing, the cost structure refers to how a service provider like Westside Energy determines the billing amounts for electricity usage. Their pricing involves a fixed base rate and variable charges based on the amount of electricity used.

• Base Rate: This is a fixed cost of $6.00 per month that each customer pays, regardless of electricity usage. It's essentially the charge just for being connected to the electric grid.


• Variable Charges: These depend on the amount of electricity used in kilowatt-hours (kWh). Westside Energy charges different rates for electricity based on usage levels, which is a common practice to encourage energy saving.

This structured approach helps manage electricity supply and demand, ensuring that customers are mindful of their energy consumption.

Analyzing Electricity Usage

Electricity usage refers to the amount of electrical energy consumed by a household or entity, typically measured in kilowatt-hours (kWh). Understanding usage is crucial as it directly influences how much one pays for their energy bill.

Here's why it's important:

• Energy Conservation: By knowing your usage, you can take steps to reduce consumption, leading to lower bills and environmental benefits.


• Billing Amounts: The cost you're charged is directly linked to how much electricity you use. In Westside Energy's model, using more than 300 kWh significantly affects the cost-calculation because of the tiered cost structure.


• Smart Management: Monitoring electricity usage can help in managing and reducing inefficient energy patterns.

Therefore, keeping track of electricity usage is key to both financial prudence and sustainability.

Graphing Functions Effectively

Graphing functions allows visual interpretation of mathematical relationships, such as cost versus electricity usage. It's a valuable tool for understanding how changes in one variable affect another.

To graph a piecewise function correctly:

• Identify Each Section: Each part of a piecewise function can represent a different scenario, such as rates changing after 300 kWh of usage.


• Plot Clearly: Draw each segment with attention to where one ends and the next begins. Here, plot a line up to 300 kWh, and another from 300 onward.


• Check Points of Interest: Calculate and mark key points where the function changes, such as at 300 kWh. For example, at 300 kWh, the cost starts transitioning from 0.10 to 0.06 additional per kWh.

This visual approach easily conveys the billing differences based on electricity usage, making it simpler to predict and compare costs.

Exploring Piecewise Functions

Piecewise functions are defined by different expressions based on the input value, making them useful for scenarios like tiered pricing. They allow formulas to change based on varying conditions.

How it applies to cost calculations:

• Multiple Ranges: For Westside Energy's billing, the function needs to handle different calculations–one formula for up to 300 kWh, and another for usage beyond that. Here, two distinct linear equations represent each tier.


• Understanding Transitions: Recognize where one function ends, and another begins, which is crucial for clarity. In this case, the cost structure switches at exactly 300 kWh.


• Simplified Representation: By using piecewise functions, utilities clearly illustrate how usage changes influence billing, promoting clarity in customer communication.

Such functions are not only mathematically efficient but also immensely practical in real-world applications like billing.