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IB Mathematics: Applications and Interpretation HL · 12.7: Mathematical Models · 12.7.6

Mixed modelling Problem set A Answers

Skills

    1. (a) k = 0.116
    2. (b) k = 0.0173
    3. (c) 0.756 years
    1. (a) A = 124, B = 16
    2. (b) k = 0.0903
    3. (c) 24.1 minutes
    1. (a) 29.5 mg
    2. (b) 15.0 hours
    1. (a) 2.19
    2. (b) For example: the rate of decrease stays proportional to the number present (nothing is added and the conditions do not change).
    3. (c) For large t the model gives a number between 0 and 1 fish (never 0), but a number of fish is a whole number and the fish would die out.

In context

    1. (a) p = 0.683, q = 2.05
    2. (b) R = 8.19
    3. (c) 74.6 ≤ A ≤ 322
    1. (a) p = 12.0
    2. (b) q = 2.00
    3. (c) 18.7 minutes
    1. (a) N = 6000e^(−0.173t)
    2. (b) After 50.2 years
    3. (c) The model never reaches 0 and gives fractions of an animal; in reality numbers are whole and chance matters when few are left.
    1. (a) k = −(1/10) ln q
    2. (b) t = 10 ln(3/14) ÷ ln q
    3. (c) 16.8 minutes

Stretch

    1. (a) a = 2.00, b = π/12 = 0.262, c = 2.00, d = 7.00
    2. (b) Q = 1.00 sin(π/12 (t − 5.00)) + 7.00
    1. (a) t = 0.0883
    2. (b) Maximum 16.8 at t = 8.98
    3. (c) Maximum (8.98, 16.8) and minimum (21.0, 11.2)
    4. (d) P + Q = 2.80 sin(π/12 (t − 2.98)) + 14.0
    1. (a) C = (2/3)e^(2k)
    2. (b) 5000 / (1 + Ce^(−6k)) = 4500
    3. (c) k = 0.448, C = 1.63
    4. (d) P = 1900
    5. (e) 9 years
    1. (a) 3800
    2. (b) A linear model grows without limit; a real population or market has a maximum size.
    3. (c) C = 1.50
    4. (d) k = 0.2
    5. (e) 4400
    1. (a) 2400
    2. (b) 198
    3. (c) t = 27.0 days
    4. (d) 12/α > 18/β

jmaths.xyz · AI HL · 12.7.6 Mixed modelling practice