Problem 34 - Algebra Form and Function

Question

Greta's Gas Company charges residential customers \(\$ 8\) per month even if they use no gas, plus \(82 \&\) per therm used. (A therm is a quantity of gas.) In addition, the company is authorized to add a rate adjustment, or surcharge, per therm. The total cost of \(g\) therms of gas is given by $$\text { Total cost }=8+0.82 g+0.109 g$$ (a) Which term represents the rate adjustment? What is the rate adjustment in cents per therm? (b) Is the expression for the total cost linear?

Step-by-Step Solution

Answer

Answer: The rate adjustment term in the formula is 0.109g, which is 10.9 cents per therm. Yes, the expression for the total cost is linear.

Step 1: Identify the rate adjustment term

The given formula for the total cost is: Total cost = \(8 + 0.82g + 0.109g\). The rate adjustment or surcharge is an additional amount that is added per therm used. Here, the terms involving "g" are the ones that depend on the number of therms used. So we have two terms: \(0.82g\) and \(0.109g\). The rate adjustment term is \(0.109g\) since it is an additional charge per therm.

Step 2: Express the rate adjustment in cents per therm

The rate adjustment in cents per therm would be \(0.109g\) for each therm used. In other words, the adjustment is \(10.9\) cents for each therm. #b)#

Step 3: Determine if the expression for total cost is linear

A linear expression has the form: \(y = mx + b\), where m and b are constants. Comparing the given total cost expression, \(8 + 0.82g + 0.109g\), with the linear expression, we can see that it matches the form: \(y = (0.82 + 0.109)g + 8\). Combining the terms with "g" gives us a total cost expression of \(y = 0.929g + 8\). Since it matches the form of a linear expression, the expression for the total cost is linear.

Key Concepts

Rate Adjustment

When discussing rate adjustment, we are talking about an extra fee that is added on a per-unit basis. In the context of Greta's Gas Company, this adjustment is applied to the amount of gas used. It serves as a surcharge that increases the cost depending on the consumption.

From the formula given in the problem, the term \(0.109g\) represents this rate adjustment. Here, "\(g\)" stands for the number of therms used. Thus, every therm of gas consumed incurs an additional cost of $0.109. To translate this into simpler terms, the rate adjustment is 10.9 cents per therm.

There are several reasons businesses might introduce a rate adjustment:

• To cover increased operational costs.
• To comply with regulatory changes.
• To maintain profitability in changing market conditions.

Understanding rate adjustments is crucial because even minor adjustments per unit can affect your total expenses significantly.

Cost Calculation

Cost calculation in this scenario involves determining how much a customer owes Greta's Gas Company based on their gas usage. The provided formula helps break it down step by step. It includes a fixed monthly charge and additional per-unit costs depending on usage.

The formula for the total cost is: \[\text{Total cost} = 8 + 0.82g + 0.109g\]This can be understood as follows:

• \(8\): A fixed monthly charge—this is constant no matter how much gas you use.
• \(0.82g\): The basic charge per therm used.
• \(0.109g\): The rate adjustment or surcharge per therm.

The goal is to combine these elements to find the complete cost. By adding the two terms that contain "\(g\)" (\(0.82g + 0.109g\)), you obtain \(0.929g\). This expression, simplified, is:\[\text{Total cost} = 0.929g + 8\]By knowing this equation, you can quickly determine the total cost for any given number of therms used, accounting for both fixed and variable elements.

Linear Equation

Linear equations play a pivotal role in simplifying cost analyses, and the exercise is a classic example. The structure of a linear equation is:\[y = mx + b\]Where:

• \(y\) is the dependent variable (total cost in this instance).
• \(m\) is the coefficient (total rate per therm).
• \(x\) is the independent variable (number of therms used \(g\)).
• \(b\) is the constant term (fixed charges, here \(8\)).

For Greta's Gas Company, we fit the total cost expression into this form: \[y = 0.929g + 8\]This ensures it is a linear equation because both "\(m\)" and "\(b\)" are constants, and the term for \(g\) (\(0.929g\)) is linear.

Linear equations are significant for their predictability and straightforward nature. They are used extensively in financial calculations because they offer a simple way to calculate costs or assess relationships between variables.