IB · Mathematics: Applications & Interpretation SL & HL · Functions

Logistic models: IB Mathematics: Applications & Interpretation questions with answers

Free Logistic models questions for IB Mathematics: Applications & Interpretation (SL & HL), HL only, part of Functions. Each question has the answer and the marking points an examiner would look for.

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Logistic models practice questions

Question 1Structured · 14 marksA* challenge

A biologist models the number of fish in a lake t years after they are introduced by N(t) = 5001 + 19e−0.4t.

(a)

Find the initial number of fish.

[1]
(b)

Find the number of fish after 3 years.

[2]
(c)

Find the time at which the population is growing fastest, and the population at that time.

[4]
(d)

An alternative model is P(t) = 25emt, which agrees with N at t = 0 and t = 3. Find m.

[3]
(e)

Compare the predictions of the two models after 10 years, and explain which model is more realistic.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)25
25 A1
1
(b)74.4
N(3) (M1)
74.4 A1
2
(c)t = 7.36, N = 250
inflexion point / maximum of N′(t) (M1)
N = half the carrying capacity (A1)
t = 7.36 A1
N = 250 A1
4
(d)m = 0.363
25e3m = 74.4 (M1)
taking logs (M1)
m = 0.363 A1
3
(e)N = 371, P = 947
N(10) = 371 A1
P(10) = 947 A1
the exponential model grows without limit R1
the logistic model is more realistic as the lake has limited resources R1
4
Question 2Structured · 6 marks

The number of people in a town who have heard a rumour, t hours after it starts, is modelled by N(t) = 50001 + 15e−0.4t.

(a)

Find the number of people who had heard the rumour initially.

[1]
(b)

State the number of people in the town, according to the model, giving a reason.

[2]
(c)

Find the time taken for 80% of the town to have heard the rumour.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)312.5
312.5 A1
1
(b)5000
as t → ∞, e−0.4t → 0 R1
5000 A1
2
(c)10.2 hours
N(t) = 4000 (M1)
attempt to solve (M1)
10.2 hours A1
3
Question 3Structured · 4 marks

The number of people who have heard a rumour is modelled by N(t) = 10001 + 19e−0.5t, t in hours.

(a)

Find the initial number of people who have heard the rumour.

[1]
(b)

Write down the long-term number predicted by the model.

[1]
(c)

Find the time when half of the long-term number have heard it.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)50
50 A1
1
(b)1000
1000 A1
1
(c)5.89 hours
solving N = 500 (M1)
5.89 A1
2
Question 4Structured · 14 marksA* challenge

A biologist models the number of fish in a lake t years after they are introduced by N(t) = 5001 + 19e−0.4t.

(a)

Find the initial number of fish.

[1]
(b)

Find the number of fish after 3 years.

[2]
(c)

Find the time at which the population is growing fastest, and the population at that time.

[4]
(d)

An alternative model is P(t) = 25emt, which agrees with N at t = 0 and t = 3. Find m.

[3]
(e)

Compare the predictions of the two models after 8 years, and explain which model is more realistic.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)25
25 A1
1
(b)74.4
N(3) (M1)
74.4 A1
2
(c)t = 7.36, N = 250
inflexion point / maximum of N′(t) (M1)
N = half the carrying capacity (A1)
t = 7.36 A1
N = 250 A1
4
(d)m = 0.363
25e3m = 74.4 (M1)
taking logs (M1)
m = 0.363 A1
3
(e)N = 282, P = 458
N(8) = 282 A1
P(8) = 458 A1
the exponential model grows without limit R1
the logistic model is more realistic as the lake has limited resources R1
4
Question 5Structured · 6 marks

The number of people in a town who have heard a rumour, t hours after it starts, is modelled by N(t) = 20001 + 15e−0.5t.

(a)

Find the number of people who had heard the rumour initially.

[1]
(b)

State the number of people in the town, according to the model, giving a reason.

[2]
(c)

Find the time taken for 80% of the town to have heard the rumour.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)125
125 A1
1
(b)2000
as t → ∞, e−0.5t → 0 R1
2000 A1
2
(c)8.19 hours
N(t) = 1600 (M1)
attempt to solve (M1)
8.19 hours A1
3
Question 6Structured · 4 marks

The number of people who have heard a rumour is modelled by N(t) = 10001 + 9e−0.5t, t in hours.

(a)

Find the initial number of people who have heard the rumour.

[1]
(b)

Write down the long-term number predicted by the model.

[1]
(c)

Find the time when half of the long-term number have heard it.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)100
100 A1
1
(b)1000
1000 A1
1
(c)4.39 hours
solving N = 500 (M1)
4.39 A1
2

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in