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A quantity of ethanol was heated until it reached boiling point - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 2
Question 10
A quantity of ethanol was heated until it reached boiling point. The temperature of the ethanol, θ °C, at time t seconds after heating began, is modelled by the equ... show full transcript
Worked Solution & Example Answer:A quantity of ethanol was heated until it reached boiling point - Edexcel - A-Level Maths Pure - Question 10 - 2020 - Paper 2
Step 1
Find A and B
Answer
To find the values of A and B:
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At t = 0, the temperature θ = 18 °C, so substituting into the equation gives:
18 = A - B$$ This simplifies to: $$A - B = 18 \, (1)$$ -
At t = 10 seconds, θ = 44 °C, substituting this value into the equation gives:
44 = A - Be^{-0.7}$$ Rearranging gives: $$A - 0.4966B = 44 \, (2)$$ (using the value of e^{-0.7} ≈ 0.4966) -
Now we can solve the equations (1) and (2) together:
Substitute into (2):
A - 8.92 + 0.4966A = 44 \ 1.4966A = 52.92 \ A = 35.39$$ 4. Now substitute A back to find B: $$B = 18 - 35.39 = -17.39$$
Thus, the complete model with A and B is:
or approximately, rounding to three significant figures:
Step 2
Evaluate the model
Answer
To evaluate the model:
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The boiling point of ethanol is approximately 78 °C.
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The maximum temperature predicted by the model occurs as t approaches infinity and can be determined from:
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Since 35.4 °C is much lower than 78 °C, the model is not appropriate for real-world conditions, as it does not reach the boiling point. Hence, this indicates that the model fails to accurately represent the behavior of ethanol when heated.