Problem 57 - Applied Calculus

Question

A utility considers demand for electricity "low" if it is below \(8 \mathrm{mkW}\) (million kilowatts), "average" if it is at least \(8 \mathrm{mkW}\) but below \(20 \mathrm{mkW},\) "high" if it is at least \(20 \mathrm{mkW}\) but below \(40 \mathrm{mkW},\) and "critical" if it is \(40 \mathrm{mkW}\) or more. Express these demand levels in interval notation. [Hint: The interval for "low" is [0,8).

Step-by-Step Solution

Answer

Low: \([0, 8)\), Average: \([8, 20)\), High: \([20, 40)\), Critical: \([40, \infty)\).

Step 1: Understand the Category Boundaries

The categories for electricity demand levels are based on certain boundaries: "low," "average," "high," and "critical." We will express these boundaries using interval notation.

Step 2: Express 'Low' Demand in Interval Notation

The 'low' demand includes all demand from 0 up to, but not including, 8 mkW. The interval notation for this range is \([0, 8)\).

Step 3: Express 'Average' Demand in Interval Notation

The 'average' demand includes all demand from 8 up to, but not including, 20 mkW. The interval notation for this range is \([8, 20)\).

Step 4: Express 'High' Demand in Interval Notation

The 'high' demand includes all demand from 20 up to, but not including, 40 mkW. The interval notation for this range is \([20, 40)\).

Step 5: Express 'Critical' Demand in Interval Notation

The 'critical' demand includes all demand from 40 mkW and above. Since there is no upper limit specified, the interval notation for this range is \([40, \infty)\).

Key Concepts

Understanding Electricity Demand Levels

Electricity demand levels are important for utilities to manage and allocate resources efficiently.
Levels like "low," "average," "high," and "critical" help in predicting and balancing power supply.
- "Low" demand is from 0 up to, but not including, 8 million kilowatts (mkW). - "Average" demand is from 8 mkW up to, but not including, 20 mkW. - "High" demand reflects consumption from 20 mkW to just below 40 mkW. - "Critical" demand covers 40 mkW and higher, a sign that consumption is at its peak. Understanding these levels helps in planning and avoiding overload situations in electricity supply systems.

Utility Demand Categorization

Categorization of utility demand is essentially breaking down power requirements into specific ranges.
This approach simplifies analyzing how electricity is consumed during different periods. An effective categorization ensures better utility management: - **Precise Resource Allocation:** Knowing the levels ensures that resources are efficiently distributed to meet demand.
- **Improved Forecasting:** By identifying these categories, utilities can predict demand patterns. - **Enhanced Infrastructure Planning:** Helps in planning infrastructure upgrades or maintenance based on demand. This structured approach makes it easier to determine when additional resources must be integrated or conserved based on clear-cut demand categories.

Piecewise Function Notation in Electricity Demand

Piecewise function notation is a powerful mathematical tool to describe functions that have different expressions over different intervals.
When applied to electricity demand, piecewise functions define rules for different demand levels.Here’s how it's applied:- For "low" demand: The function might be represented as \( f(x) = x \) for \( x \) in \[0, 8)\].
- "Average" demand could be defined as \( f(x) = x \) for \( x \) in \[8, 20)\].- "High" demand is \( f(x) = x \) capturing demand from \[20, 40)\].
- Finally, "critical" demand as \( f(x) = x \) for \( x \) in \[40, \infty)\].Using piecewise functions can clearly depict situations where demand varies greatly across different intervals, providing utilities with critical insights for better decision-making.