At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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!
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.