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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
To express the equation 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{Hence, we can identify:}- m = b
- c = log₁₀a.
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 in Figure 3, we can determine the gradient and intercept. The estimated values are:
- The gradient (b) can be approximated by looking at the slope, which is around 2.3.
- The value of a can be determined using the model:
Assuming a ≈ 10^6.3, we can calculate:
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 between log₁₀N and log₁₀T is only valid for values of log₁₀N between 1 and 4.5, as shown in the graph. Thus, estimating the day for which N exceeds 1,000,000 (log₁₀N = 6) cannot be reliably determined because extrapolation beyond the provided data points is not certain. Therefore, any estimates might not accurately reflect the actual growth curve of microbes.
Step 4