*Globalpetrolprices.com is a website that compares utility bills for households and businesses. It has been observed that Denmark, German and Italy have the highest household electricity costs. Suppose Italy’s electricity pricing is as follows: Each household bears a minimum of 50$ fixed charge and an additional 0.78$ for each unit of electricity consumed by the household.*

Based on the above scenario, please answer the following instructions:

(i) Formulate a linear function for the electricity pricing based on the consumption. Please clearly define each variable.

(ii) How does the average rate of change in electricity price with consumption impact a consumer's monthly bill? Answer by calculating the average rate of change.


Let's address each part of your request step by step.

### (i) Formulating a Linear Function for Electricity Pricing

To model Italy's electricity pricing as a linear function, we'll define the variables and construct the function based on the given dataLet's address each part of your request step by step.

### (i) Formulating a Linear Function for Electricity Pricing

To model Italy's electricity pricing as a linear function, we'll define the variables and construct the function based on the given 

- Let $ x $ be the number of units of electricity consumed by a household.
- The fixed charge (base cost) is $50, which every household has to pay.
- The variable cost is $0.78 per unit of electricity consumed.

The total cost $ C(x) $, based on the number of units consumed $ x $, is a linear function of $ x $. The function is:

\[
C(x) = 0.78x + 50
\]

Where:
- $ C(x) $ represents the total cost in dollars for consuming $ x $ units of electricity.
- $ x $ is the number of units of electricity consumed.
- $50 is the fixed charge.
- $0.78 is the cost per unit of electricity.

### (ii) Average Rate of Change in Electricity Price

The average rate of change of the electricity cost function measures how the cost changes with each additional unit of electricity consumed.

In the function $ C(x) = 0.78x + 50 $, the term that affects the average rate of change with respect to $ x $ is the coefficient of $ x $, which is $0.78. This means that for every additional unit of electricity consumed, the total cost increases by $0.78.

#### Impact on Consumer's Monthly Bill:

Since the average rate of change is constant (0.78), the impact is that for each additional unit of electricity consumed, the bill will rise by $0.78. This linear relationship indicates that the more a household consumes, the more they will pay, but the cost per unit remains steady at $0.78.

### Summary:
- **Linear Function:** $ C(x) = 0.78x + 50 $
- **Average Rate of Change:** For each additional unit of electricity, the bill increases by $0.78. This directly impacts the monthly bill in a proportional way, with no change in the rate regardless of the number of units consumed.

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Would you like more details or have any questions on this?

Here are five related questions:
1. How would the equation change if the fixed charge increased?
2. What would the cost be if a household consumed 150 units of electricity?
3. Can this linear model be applied if there were discounts for higher consumption?
4. What if the cost per unit decreased after a threshold (e.g., for consumption above 200 units)?
5. How would a comparison between two different households' consumption look like using this model?

**Tip:** Always check for fixed charges in utility bills, as they can significantly affect total costs even with low usage.

Globalpetrolprices.com is a website that compares utility bills for households and businesses. It has been observed that Denmark, German and Italy have the highest household electricity costs. Suppose Italy’s electricity pricing is as follows: Each household bears a minimum of 50$ fixed charge and an additional 0.78$ for each unit of electricity consumed by the household.

Learn how to formulate a linear function for Italy's electricity pricing, where each household pays a fixed charge of $50 and $0.78 per unit of electricity consumed. This problem explores key concepts in algebra, such as linear functions and the average rate of change. Suitable for students in Grades 8-10.

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