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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 13 - 2017 - Paper 1
Question 13
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 13 - 2017 - Paper 1
Step 1
Show that this relationship can be expressed in the form log₁₀N = m log₁₀T + c
Answer
Starting from the equation ( N = aT^b ), we take the logarithm base 10 of both sides:
Using the property of logarithms, this can be expressed as:
This rearranges to:
Thus, we define ( m = b ) and ( c = \log_{10}a ).
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
From the graph (Figure 3), we can identify the coordinates for ( T = 3 ) days. By assuming the line of best fit suggests:
( \log_{10}N \approx 2.3 + 0.63 \cdot \log_{10}3 )
Using ( \log_{10}3 \approx 0.477 ), we calculate:
Thus, finding ( N ) gives us:
So, the estimated number of microbes is approximately 800.
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
For ( N = 1,000,000 ), we have:
The graph provides values for ( \log_{10}N ) between 1.8 and 4.5. However, it does not extend to 6, therefore we cannot be certain about the exact point at which ( N ) exceeds 1,000,000. We can only interpolate within the recorded limits.
Step 4
With reference to the model, interpret the value of the constant a.
Answer
In the model ( N = a T^b ), the constant ( a ) represents the initial number of microbes present in the culture at time ( T = 1 ) day. Specifically, when ( T = 1 ):
Thus, ( a ) gives the baseline microbial population before the growth described by the power law begins.