Answer :
Certainly! Let's work through this problem step-by-step.
### Part (i): Expected Summer Bill Calculation
Given information:
- In winter, the average temperature was [tex]\(8^\circ\)[/tex]C, and the bill was £365.
- The temperature-Cost (C) relationship is inverse, meaning that as the temperature increases, the cost decreases proportionally.
To calculate the expected bill for the summer when the average temperature is [tex]\(20^\circ\)[/tex]C, we can use the concept of inverse variation. Inverse variation implies that [tex]\( C \times T = k \)[/tex], where [tex]\(k\)[/tex] is a constant.
From winter data, we can calculate this constant:
[tex]\[ k = C_{\text{winter}} \times T_{\text{winter}} \][/tex]
[tex]\[ k = 365 \text{ pounds} \times 8^\circ \text{C} \][/tex]
[tex]\[ k = 2920 \text{ pounds} \cdot \text{degree Celsius} \][/tex]
Now, we use this constant to find the expected summer bill [tex]\(C_{\text{summer}}\)[/tex] when the temperature is [tex]\(20^\circ\)[/tex]C:
[tex]\[ C_{\text{summer}} = \frac{k}{T_{\text{summer}}} \][/tex]
[tex]\[ C_{\text{summer}} = \frac{2920 \text{ pounds} \cdot \text{degree Celsius}}{20^\circ \text{C}} \][/tex]
[tex]\[ C_{\text{summer}} = 146 \text{ pounds} \][/tex]
Therefore, I would expect to pay £146 in the summer.
### Part (ii): Graph Sketch
While it is not possible to sketch the graph directly in text, I'll describe it for you:
- Axes: The horizontal axis (x-axis) represents the temperature [tex]\(T\)[/tex] in degrees Celsius. The vertical axis (y-axis) represents the cost [tex]\(C\)[/tex] in pounds.
- Curve: The relationship is a form of a rectangular hyperbola. The curve starts at a high value on the y-axis when the temperature is low (indicating high cost) and gradually curves downwards, approaching the x-axis as the temperature increases (indicating lower cost).
The hyperbolic nature shows that as temperature increases, the cost decreases proportionally, and this relationship is continuous and smooth.
### Part (iii): Realism of the Model
The given model assumes a perfect inverse relationship between temperature and electricity cost. While this can highlight a general trend, the model may not be completely realistic due to several factors:
1. Non-linearity: Electricity costs are likely influenced by numerous factors, not just temperature. This includes household usage patterns, insulation efficiency, the efficiency of heating/cooling systems, and electricity rates.
2. Fixed Costs: Some fixed costs in the electricity bill do not vary with temperature.
3. Seasonal Variations: Different appliances (heating in winter vs. cooling in summer) have different energy efficiencies, potentially affecting costs differently across seasons.
In summary, while the model provides a useful approximation and highlights the general inverse relationship, it oversimplifies the multiple factors impacting electricity bills, and thus, may not fully capture the actual dynamics of electricity usage and cost.
I hope this gives you a clear understanding of the inverse relationship and the associated electricity costs across different temperature conditions!
### Part (i): Expected Summer Bill Calculation
Given information:
- In winter, the average temperature was [tex]\(8^\circ\)[/tex]C, and the bill was £365.
- The temperature-Cost (C) relationship is inverse, meaning that as the temperature increases, the cost decreases proportionally.
To calculate the expected bill for the summer when the average temperature is [tex]\(20^\circ\)[/tex]C, we can use the concept of inverse variation. Inverse variation implies that [tex]\( C \times T = k \)[/tex], where [tex]\(k\)[/tex] is a constant.
From winter data, we can calculate this constant:
[tex]\[ k = C_{\text{winter}} \times T_{\text{winter}} \][/tex]
[tex]\[ k = 365 \text{ pounds} \times 8^\circ \text{C} \][/tex]
[tex]\[ k = 2920 \text{ pounds} \cdot \text{degree Celsius} \][/tex]
Now, we use this constant to find the expected summer bill [tex]\(C_{\text{summer}}\)[/tex] when the temperature is [tex]\(20^\circ\)[/tex]C:
[tex]\[ C_{\text{summer}} = \frac{k}{T_{\text{summer}}} \][/tex]
[tex]\[ C_{\text{summer}} = \frac{2920 \text{ pounds} \cdot \text{degree Celsius}}{20^\circ \text{C}} \][/tex]
[tex]\[ C_{\text{summer}} = 146 \text{ pounds} \][/tex]
Therefore, I would expect to pay £146 in the summer.
### Part (ii): Graph Sketch
While it is not possible to sketch the graph directly in text, I'll describe it for you:
- Axes: The horizontal axis (x-axis) represents the temperature [tex]\(T\)[/tex] in degrees Celsius. The vertical axis (y-axis) represents the cost [tex]\(C\)[/tex] in pounds.
- Curve: The relationship is a form of a rectangular hyperbola. The curve starts at a high value on the y-axis when the temperature is low (indicating high cost) and gradually curves downwards, approaching the x-axis as the temperature increases (indicating lower cost).
The hyperbolic nature shows that as temperature increases, the cost decreases proportionally, and this relationship is continuous and smooth.
### Part (iii): Realism of the Model
The given model assumes a perfect inverse relationship between temperature and electricity cost. While this can highlight a general trend, the model may not be completely realistic due to several factors:
1. Non-linearity: Electricity costs are likely influenced by numerous factors, not just temperature. This includes household usage patterns, insulation efficiency, the efficiency of heating/cooling systems, and electricity rates.
2. Fixed Costs: Some fixed costs in the electricity bill do not vary with temperature.
3. Seasonal Variations: Different appliances (heating in winter vs. cooling in summer) have different energy efficiencies, potentially affecting costs differently across seasons.
In summary, while the model provides a useful approximation and highlights the general inverse relationship, it oversimplifies the multiple factors impacting electricity bills, and thus, may not fully capture the actual dynamics of electricity usage and cost.
I hope this gives you a clear understanding of the inverse relationship and the associated electricity costs across different temperature conditions!
