A population of mosquitoes decreases exponentially. The size of the population, \(P\) , after \(t\) days is modelled by

\(P = 3200 \times {2^{ - t}} + 50\) , where \(t \geqslant 0\) .

Write down the exact size of the initial population.

Find the size of the population after \(4\) days.

Calculate the time it will take for the size of the population to decrease to \(60\).

The population will stabilize when it reaches a size of \(k\) .

Write down the value of \(k\) .

\(3250\)        (A1)     (C1)

\(3200 \times {2^{ - 4}} + 50\)        (M1)

Note: Award (M1) for substituting \(t\) into exponential equation.

\( = 250\)        (A1)    (C2)

\(3200 \times {2^{ - t}} + 50 = 60\)       (M1)

Note: Award (M1) for setting up the equation used in part (b).

OR

(M1)

Note: Award (M1) for a decreasing exponential graph intersecting a horizontal line.

\((t = )\,\,8.32\,\,(8.32192...)\) (days)        (A1)     (C2)

Note: Accept a final answer of “\(8\) days, \(7\) hours and \(44\) minutes”, or equivalent. Award (M0)(A0) for an answer of \(8\) days with no working

\(50\)       (A1)     (C1)

Question 13: Exponential model.

Most candidates were able to correctly substitute values into the given exponential model but only the stronger ones found a correct answer. It was expected that candidates would use their calculator to solve the exponential equation rather than use logarithms which is not in the syllabus. The concept of the population stabilizing (horizontal asymptote) was not widely understood.

Question 13: Exponential model.
Most candidates were able to correctly substitute values into the given exponential model but only the stronger ones found a correct answer. It was expected that candidates would use their calculator to solve the exponential equation rather than use logarithms which is not in the syllabus. The concept of the population stabilizing (horizontal asymptote) was not widely understood.

Question 13: Exponential model.

Most candidates were able to correctly substitute values into the given exponential model but only the stronger ones found a correct answer. It was expected that candidates would use their calculator to solve the exponential equation rather than use logarithms which is not in the syllabus. The concept of the population stabilizing (horizontal asymptote) was not widely understood.

Question 13: Exponential model.

Most candidates were able to correctly substitute values into the given exponential model but only the stronger ones found a correct answer. It was expected that candidates would use their calculator to solve the exponential equation rather than use logarithms which is not in the syllabus. The concept of the population stabilizing (horizontal asymptote) was not widely understood.