Problem 68 - Algebra and Trigonometry with Analytic Geometry

Question

An electric company charges its customers \(\$ 0.0577\) per kilowatt-hour \((\mathrm{kWh})\) for the first \(1000 \mathrm{kWh}\) used, \(\$ 0.0532\) for the next \(4000 \mathrm{kWh}\), and \(\$ 0.0511\) for any \(\mathrm{kWh}\) over 5000 . Find a piecewise-defined function \(C\) for a customer's bill of \(x \mathrm{kWh}\).

Step-by-Step Solution

Answer

The cost function is: \( C(x) = 0.0577x \) for \( 0 \leq x \leq 1000 \); \( C(x) = 57.7 + 0.0532(x - 1000) \) for \( 1000 < x \leq 5000 \); \( C(x) = 270.5 + 0.0511(x - 5000) \) for \( x > 5000 \).

Step 1: Determine the Billing Tiers

The electric company has three tiers for billing based on kilowatt-hours (kWh) used. These are: the first tier for the first 1000 kWh charged at $0.0577 per kWh, the second tier for the next 4000 kWh (more than 1000 kWh and up to 5000 kWh) charged at $0.0532 per kWh, and the third tier for usage over 5000 kWh charged at $0.0511 per kWh.

Step 2: Define the Function for the First 1000 kWh

For usage up to 1000 kWh, the cost function is simply the number of kWh multiplied by $0.0577. Thus, for \(0 \leq x \leq 1000\), the cost function is: \[ C(x) = 0.0577x \]

Step 3: Define the Function for the Next 4000 kWh

For usage more than 1000 kWh and up to 5000 kWh, the cost includes both the first 1000 kWh and the additional kWh in this range. The first 1000 kWh cost is fixed at \(0.0577 \times 1000 = 57.7\). For the remaining \((x - 1000)\) kWh up to 5000 kWh, the cost is \(0.0532\) per kWh. Thus, for \(1000 < x \leq 5000\), the function is: \[ C(x) = 57.7 + 0.0532(x - 1000) \]

Step 4: Define the Function for Over 5000 kWh

For usage over 5000 kWh, the total includes the costs for the first 5000 kWh and the remaining usage above 5000. The cost for the first 5000 kWh is \(57.7 + 0.0532 \times 4000 = 270.5\). For the remaining \((x - 5000)\) kWh, the cost is \(0.0511\) per kWh. Thus, for \(x > 5000\), the function is: \[ C(x) = 270.5 + 0.0511(x - 5000) \]

Step 5: Combine into a Piecewise Function

Now, combine the three distinct cost calculations into a single piecewise function for the entire range of electricity usage: \[ C(x) = \begin{cases} 0.0577x, & 0 \leq x \leq 1000 \ 57.7 + 0.0532(x - 1000), & 1000 < x \leq 5000 \ 270.5 + 0.0511(x - 5000), & x > 5000 \end{cases} \]

Key Concepts

Billing Tiers

In the context of calculating electricity costs, billing tiers refer to the different price categories applied as your usage increases. Electricity companies often use these tiers to encourage energy conservation by charging different rates based on consumption thresholds.
With the electric company in our example, they organize their pricing into three billing tiers:

• First tier: For the initial 1000 kilowatt-hours (kWh) of usage, customers are billed at a rate of $0.0577 per kWh.
• Second tier: For additional usage from 1001 to 5000 kWh, the rate drops slightly to $0.0532 per kWh.
• Third tier: Usage beyond 5000 kWh is billed at an even lower rate of $0.0511 per kWh.

Each tier is designed to reflect the cost structure of providing electricity, aimed at balancing the operational costs with consumer demand. Understanding these tiers helps customers predict and manage their monthly bills.

Electricity Cost

Electricity cost is the expense incurred for consuming electrical energy, measured per kilowatt-hour (kWh). This cost is reflected in your electricity bills and varies based on consumption levels along with the billing tiers applied by your electric company.
To compute your electricity cost, you need to know:

• The total amount of kWh you've used within the billing period.
• The specific rates associated with each portion of your usage. As you consume more energy, you move into different tiers with varied prices.

For instance, to calculate how much you'll pay for using 2000 kWh: - The first 1000 kWh are charged at the first tier rate. - The next 1000 kWh fall into the second tier.
Each increment in kWh should be meticulously accounted for using the appropriate tier rate, allowing you to precisely forecast your costs.

Function Definition

Creating a function to calculate electricity costs involves defining a piecewise function. This type of function uses distinct formulas for different sections of its domain, perfectly fitting the scenario of tiered billing.
For the task at hand, the function is defined as:

• For 0 to 1000 kWh: \[ C(x) = 0.0577x \]Here, the cost is simply calculated by multiplying the usage in this range by the rate for the first tier.
• For 1001 to 5000 kWh:\[ C(x) = 57.7 + 0.0532(x - 1000) \]The first 1000 kWh are fixed, and you add the cost of additional usage up to 5000 kWh using the second tier rate.
• For over 5000 kWh:\[ C(x) = 270.5 + 0.0511(x - 5000) \]Here, the fixed cost for the first 5000 kWh is combined with the additional cost past this point using the third tier rate.

Defining the function piecewise allows a tailored calculation applied to the distinct ranges of usage, ensuring accuracy.

Step by Step Solution

Understanding the piecewise function through this example involves breaking it down into easy steps.**Step 1: Recognize Billing Tiers**
The first step is acknowledging the tiered structure that the electric company offers based on usage levels. This tier recognition frames the rest of your calculations.**Step 2: Define for First Tier (0-1000 kWh)**
This initial section of the function: \[ C(x) = 0.0577x \]directly applies the tier price to your consumption within this limit.**Step 3: Define for Second Tier (1001-5000 kWh)**
Here, calculate the fixed cost for 1000 kWh, and then apply the next rate to usage beyond 1000 kWh with:\[ C(x) = 57.7 + 0.0532(x - 1000) \]**Step 4: Define for Third Tier (Over 5000 kWh)**
Calculate up to 5000 kWh and then adjust for usage above, using:\[ C(x) = 270.5 + 0.0511(x - 5000) \]This methodical layer shows how to choose values and applies the rules step by step, bringing intricate calculations into simpler terms. Each piece of the function is tailored to a specific usage range but collectively provides a complete billing solution.