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In a simple model, the value, $V_t$, of a car depends on its age, $t$, in years - Edexcel - A-Level Maths: Pure - Question 9 - 2019 - Paper 2
Question 9
In a simple model, the value, $V_t$, of a car depends on its age, $t$, in years. The following information is available for car A - its value when new is £20000 - ... show full transcript
Worked Solution & Example Answer:In a simple model, the value, $V_t$, of a car depends on its age, $t$, in years - Edexcel - A-Level Maths: Pure - Question 9 - 2019 - Paper 2
Step 1
Use an exponential model to form, for car A, a possible equation linking $V_t$ with $t$.
Answer
To model the depreciation of car A, we start with the exponential decay formula:
Where:
- is the initial value (£20000)
- is the value after years
- is the decay constant.
Using the information that the value after one year is £16000:
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Substitute and :
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Solve for :
e^{-k} = rac{16000}{20000} = 0.8
Taking natural logarithm:
Thus, .
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Finally, the model becomes:
Step 2
Evaluate the reliability of your model in light of this information.
Answer
After 10 years, the model predicts:
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Substitute into the model:
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Calculate:
This is significantly higher than the observed value of £2000.
Therefore, while the model provides a close estimate, it may not be entirely reliable, as it underestimates the depreciation in value over a longer period.
Step 3
Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B.
Answer
To adjust the equation for car B, which depreciates more slowly than car A, we can modify the decay constant to a lower value.
For example, if car B depreciates more gently, we could set to a value like or to reflect reduced depreciation.
The new model would take the form:
with a suitable .