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In a controlled experiment, the number of microbes, N, present in a culture T days after the start of the experiment were counted - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 1
Question 14
In a controlled experiment, the number of microbes, N, present in a culture T days after the start of the experiment were counted. N and T are expected to satisfy a... show full transcript
Worked Solution & Example Answer:In a controlled experiment, the number of microbes, N, present in a culture T days after the start of the experiment were counted - Edexcel - A-Level Maths Pure - Question 14 - 2017 - Paper 1
Step 1
Show that this relationship can be expressed in the form log₁₀N = m log₁₀T + c
Answer
To express the relationship N = aT^b in logarithmic form, take the logarithm of both sides:
egin{align*} ext{log}_ ext{10}N &= ext{log}_ ext{10}(aT^b) \\ &= ext{log}_ ext{10}a + ext{log}_ ext{10}(T^b) \\ &= ext{log}_ ext{10}a + b ext{log}_ ext{10}T \\ ext{Therefore, } m &= b ext{ and } c = ext{log}_ ext{10}a. ext{This shows the required form.} \end{align*}Step 2
Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
Answer
Using the graph from Figure 3, we can estimate the gradient (b) and y-intercept (c).
Assume the gradient (b) is approximately 2.3 and intercept (a) is around 10^6.3 based on analysis.
To find N when T = 3:
Using the model:
This gives us an approximate number of microbes around 800, with an acceptance range of ±150.
Step 3
Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1,000,000.
Answer
The relationship used is limited to the range of T values from 0 to 2.5 in the provided data. Extrapolating beyond this range is not reliable, as we can only estimate values between 1.8 and 4.5 and cannot infer behavior outside this data set.
Step 4