Problem 31 - Algebra Form and Function

Question

Long Island Power Authority charges its residential customers a monthly service charge plus an energy charge based on the amount of electricity used. \({ }^{2}\) The monthly cost of electricity is approximated by the function: \(C=f(h)=36.60+0.14 h,\) where \(h\) represents the number of kilowatt hours \((\mathrm{kWh})\) of electricity used in excess of \(250 \mathrm{kWh}\). (a) What does the coefficient 0.14 mean in terms of the cost of electricity? (b) Find \(f(50)\) and interpret its meaning.

Step-by-Step Solution

Answer

Answer: The coefficient 0.14 represents the cost of electricity per excess kilowatt-hour. When a residential customer consumes 50 additional kilowatt-hours, the total cost of electricity would be $43.60.

Step 1: (a) Understanding the meaning of the coefficient 0.14)

The given function \(C=f(h)=36.60+0.14h\) shows the relation between the cost \((C)\) and the number of excess kilowatt-hours \((h)\). Here, the coefficient 0.14 is the rate at which the cost of electricity is changing relative to the excess kilowatt-hours used, or in simpler words, it is the cost of the electricity per excess kilowatt-hour.

Step 2: (b) Finding and interpreting the value of f(50))

Find the cost of electricity when 50 additional kilowatt-hours are used by calculating the value of the function at \(h=50\): \(f(50) = 36.60 + 0.14(50)\) \(f(50) = 36.60 + 7\) \(f(50) = 43.60\) The value of the function at \(h=50\) is 43.60. This means that the total cost of electricity for a residential customer when consuming 50 additional kilowatt-hours, i.e., 300 kWh, would be $43.60.

Key Concepts

Understanding Linear Functions

A linear function is a type of function that is very common in algebra. It's represented by a straight line on a graph. In mathematical terms, a linear function can be written as \( y = mx + c \), where:

• \( y \) represents the dependent variable (output).
• \( x \) represents the independent variable (input).
• \( m \) is the slope of the line, showing the rate of change.
• \( c \) is the y-intercept, the point where the line crosses the y-axis.

The simplicity of linear functions makes them easy to work with and useful for modeling real-world situations. For example, in the given exercise, the cost function is a linear function where the total cost of electricity is calculated based on the number of kilowatt-hours used. This relationship is linear because the cost increases at a constant rate for each additional kilowatt-hour.

Exploring Cost Functions

A cost function is a specific form of linear function that describes how a cost is accumulated or distributed. In the context of the Long Island Power Authority's billing for electricity, the cost function \( C(f(h)) = 36.60 + 0.14h \) clearly outlines:

• The fixed monthly charge of $36.60, representing the basic service charge.
• An additional charge for every excess kilowatt-hour used, depicted by the term \( 0.14h \).

Cost functions are invaluable in economic contexts as they allow consumers and businesses to predict expenses based on varying inputs or activity levels. By relying on a simple equation, customers can estimate their bills by knowing their usage patterns, helping them budget better and make informed decisions on energy consumption.

Interpretation of Coefficients in Context

In algebraic functions, coefficients play a crucial role in giving clear and quantifiable meaning to the variables they accompany. In the exercise, the coefficient 0.14 in the equation \( C = 36.60 + 0.14h \) is particularly important because it tells us how much the cost increases per excess kilowatt-hour of electricity used. This means:

• For each extra kilowatt-hour beyond the included 250 kWh, an additional 14 cents is added to the customer's bill.
• This constant rate provides clarity on the cost structure and aids in estimating potential expenses with accuracy.

Understanding coefficients helps in breaking down complex situations into simple, interpretable parts. It makes it easier to grasp how each component of an equation contributes to the final outcome, allowing for strategic planning and analysis.