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A scientist is studying the growth of two different populations of bacteria - Edexcel - A-Level Maths Pure - Question 10 - 2021 - Paper 1
Question 10
A scientist is studying the growth of two different populations of bacteria. The number of bacteria, N, in the first population is modelled by the equation N = Ae^... show full transcript
Worked Solution & Example Answer:A scientist is studying the growth of two different populations of bacteria - Edexcel - A-Level Maths Pure - Question 10 - 2021 - Paper 1
Step 1
Find a complete equation for the model.
Answer
To derive the complete equation for the first population, we start with the general formula:
Given that at time r = 0, N = 1000, we have:
Which simplifies to:
Next, we know that it takes 5 hours for the population to double, therefore:
Dividing both sides by 1000 yields:
Taking the natural logarithm of both sides gives:
Solving for k results in:
k = rac{ ext{ln}(2)}{5}
So, substituting A and k back into the equation, we have:
N = 1000e^{ rac{ ext{ln}(2)}{5} r}
This is the complete equation for the model.
Step 2
Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study.
Answer
To find the rate of increase, we first differentiate the equation with respect to r:
rac{dN}{dr} = 1000 rac{ln(2)}{5} e^{ rac{ln(2)}{5} r}
Now substituting r = 8 into this derivative gives:
rac{dN}{dr} = 1000 rac{ln(2)}{5} e^{ rac{ln(2)}{5} imes 8}
Calculating this produces a value. To express this accurately, we calculate the constant first:
- Approximate value of :
- Replace it in the equation:
rac{dN}{dr} = 1000 imes rac{0.693}{5} e^{ rac{0.693}{5} imes 8}
Evaluating further yields the result:
Approximately 420 bacteria per hour (to 2 significant figures).
Step 3
Find the value of T.
Answer
To find T, we equate the two population models at T hours:
1000e^{ rac{ ext{ln}(2)}{5} T} = 500e^{kT}
Substituting our earlier value for k:
1000e^{ rac{ ext{ln}(2)}{5} T} = 500e^{ rac{ ext{ln}(2)}{5} T}
Dividing both sides by 500 yields:
2e^{ rac{ ext{ln}(2)}{5} T} = e^{ rac{ ext{ln}(2)}{5} T}
This simplifies down to:
2 = e^{( rac{ ext{ln}(2)}{5} - k) T}
Taking the natural logarithm results in:
T = rac{12.5}{ rac{ ext{ln}(2)}{5}} = 12.5 ext{ hours}
So, the value for T is 12.5 hours.