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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Studies |
While studying a new disease, scientists found that the number of toxic cells in the bloodstream increased over time, according to the model \(D(t)=12×(1.16)^t , t \ge 0\) where \(D\) is the number of the toxic cells in the bloodstream per litre and \(t\) is the time in hours.
(a) Find the number of toxic cells in the bloodstream at \(t=0\).
(b) Calculate the number of toxic cells in the bloodstream after 3 hours.
(c) Determine the time it takes for the number of toxic cells in the bloodstream to first exceed to 200 per litre. Give your answer to the nearest minute.
2. | IB Standard |
Percy Cod and Fran Finklestein are both researchers working at different universities. They are each studying a different colony of bacteria which coincidentally start increasing in size at the same time.
The number of bacteria in Percy's colony, after \(t\) hours, is modelled by the function \(P(t)=8e^{0.3t}\).
(a) Find the initial number of bacteria in Percy's colony.
(b) Find the number of bacteria in Percy's colony after four hours.
(c) How long does it take for the number of bacteria in Percy's colony to reach 350?
The number of bacteria in Fran's colony, after t hours, is modelled by the function \(F(t)=16e^{kt}\).
(d) After four hours, there are 35 bacteria in Fran's colony. Find the value of \(k\).
(e) The number of bacteria in Percy's colony first exceeds the number of bacteria in Fran's colony after \(n\) hours, where \(n\in \mathbb Z\). Find the value of \(n\).
3. | A-Level |
In a remote lake it was noticed by conservationists that a disease was rapidly spreading amongst two species of fish, R and S, which is reducing their numbers. The conservationists calculated that the numbers of each type of fish can be modelled by the functions:
$$ r(t) = 9000e^{-\frac{1}{10}t} $$and
$$ s(t) = 6000e^{-\frac{1}{20}t} $$respectively where t is the time in weeks after the disease was first detected on the 2nd August 2019.
(a) Use the two models to find the number of species R and S on 2nd August 2019.
(b) Find the number of species S after 24 weeks from 2nd August 2019, giving your answer to the nearest 10.
(c) After how many whole weeks will the number of species R first fall below 4500?
(d) Use logarithms and the two models to calculate the value of t when the number of species S will be three times that of species R. Give your answer to the nearest whole number.
(e) When \(t = T\) the number of species \(S\) first exceeds that of species R by 500. Use this information and the two models to derive a quadratic equation in \(x\) where:
$$ x=e^{-\frac{1}{20}T} $$(f) Hence find the number of days after 2nd August 2019 when this difference of 500 fish will first occur. Give your answer to the nearest day.
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