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

Mixed modelling Problem set B Answers

Skills

    1. (a) k = 0.0495
    2. (b) k = 0.0277
    3. (c) 1.18 years
    1. (a) A = −19, B = 23
    2. (b) k = 0.0346
    3. (c) 53.3 minutes
    1. (a) 150 g
    2. (b) 39.8 days
    1. (a) 3.73
    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 bacteria (never 0), but a number of bacteria is a whole number and the bacteria would die out.

In context

    1. (a) p = 10.6, q = 27.7
    2. (b) R = 61.7
    3. (c) 8.25 ≤ A ≤ 34.0
    1. (a) p = 15.0
    2. (b) q = 3.00
    3. (c) 17.3 minutes
    1. (a) N = 900e^(−0.116t)
    2. (b) After 58.9 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 = −(1/15) ln q
    2. (b) t = 15 ln(19/84) ÷ ln q
    3. (c) 32.2 minutes

Stretch

    1. (a) a = 2.00, b = π/12 = 0.262, c = 3.24, d = 4.50
    2. (b) Q = 1.00 sin(π/12 (t − 7.24)) + 4.50
    1. (a) t = 1.24
    2. (b) Maximum 11.6 at t = 10.5
    3. (c) Maximum (10.5, 11.6) and minimum (22.5, 6.35)
    4. (d) P + Q = 2.65 sin(π/12 (t − 4.51)) + 9.00
    1. (a) C = (1/4)e^(3k)
    2. (b) 800 / (1 + Ce^(−7k)) = 760
    3. (c) k = 0.390, C = 0.804
    4. (d) P = 443
    5. (e) 11 months
    1. (a) 1520
    2. (b) A linear model grows without limit; a real population or market has a maximum size.
    3. (c) C = 2.75
    4. (d) k = 0.3
    5. (e) 2640
    1. (a) 2400
    2. (b) 756
    3. (c) t = 44.0 hours
    4. (d) 20/α > 10/β

jmaths.xyz · AI HL · 12.7.6 Mixed modelling practice