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Theresa bought a house on 2 January 1970 for £8000 - AQA - A-Level Maths: Pure - Question 8 - 2019 - Paper 2
Question 8
Theresa bought a house on 2 January 1970 for £8000. The house was valued by a local estate agent on the same date every 10 years up to 2010. The valuations are sho... show full transcript
Worked Solution & Example Answer:Theresa bought a house on 2 January 1970 for £8000 - AQA - A-Level Maths: Pure - Question 8 - 2019 - Paper 2
Step 1
Show that $V = pq$ can be written as $ ext{log}_{10} V = ext{log}_{10} p + ext{log}_{10} q$
Answer
To manipulate the equation , we first take the logarithm (base 10) of both sides:
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Starting with the equation:
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Apply the logarithm:
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Using the property of logarithms that states , we can rewrite the equation:
Thus, we have shown that can indeed be expressed as .
Step 2
The values in the table of log$_{10} V$ against t have been plotted and a line of best fit has been drawn on the graph below.
Answer
From the graph, we can analyze the trend of the data points plotted:
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Identify points: The plotted data points correspond to the following pairs:
- (0, 3.90)
- (10, 4.28)
- (20, 4.63)
- (30, 4.91)
- (40, 5.31)
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Calculate the slope (gradient) of the line of best fit:
- Using two points, for example, (0, 3.90) and (40, 5.31):
- Slope = rac{5.31 - 3.90}{40 - 0} = rac{1.41}{40} = 0.03525
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Considering logarithmic relationships, we can establish the equation for the line of best fit. If the line follows the form of (where is the slope and is the y-intercept), we can deduce the values accordingly and derive constants p and q for further calculations.