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IB Mathematics: Applications and Interpretation HL · 12.7: Mathematical Models · 12.7.2

Name Date

Exponential models, half-life and logarithmic models Problem set A

Show your working. Use the calculator for equations and for a and b. Give answers to 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. The mass of a sample is \(M=M_0e^{-0.035t}\), with t in hours. Find the half-life.

  2. Starting from \(y=y_0e^{kt}\), where k < 0, show that the half-life is \(-\dfrac{\ln2}{k}\).

  3. A liquid is at 90°C in a room at 30°C. The difference from the room temperature halves every 8 minutes. Write the temperature as \(T=d+Ae^{kt}\), with t in minutes.

  4. A model \(y=a+b\ln x\) passes through (2, 3) and (20, 9). Find a and b.

  5. A model is \(y=2.4+3.1\ln x\).

    1. (a) Find y when x = 12.
    2. (b) Find x when y = 15.

In context

  1. A radioactive sample is placed in a detector. The count rate is 200 counts per minute at t = 0, where t is in minutes. The room has a background count of 40 counts per minute. The count above the background halves every 30 minutes.

    1. (a) Find a model \(R=d+Ae^{kt}\) for the count rate.
    2. (b) Find when the count rate falls to 55 counts per minute.

  2. A drug has concentration 12 mg/l in the blood at t = 0, where t is in hours. The concentration halves every 6 hours.

    1. (a) Write a model \(C=Ae^{kt}\) for the concentration.
    2. (b) Find the concentration after 15 hours.

  3. The height h metres of a young tree at age t years is modelled by \(h=a+b\ln t\). The tree is 1.1 m tall at 2 years and 3.4 m tall at 10 years.

    1. (a) Find a and b.
    2. (b) Find the height at 6 years.
    3. (c) Find the age at which the tree is 5 m tall.

  4. The number of cases of an infection grows by 8% each day. It is modelled by \(N=N_0e^{kt}\), where t is in days.

    1. (a) Find k.
    2. (b) Find the time for the number of cases to triple.

  5. A wooden beam from an old barn has 45% of the carbon-14 of living wood. The half-life of carbon-14 is 5730 years. The amount is \(M=M_0e^{kt}\), where t is in years.

    1. (a) Find k.
    2. (b) Find the age of the beam.

Stretch

  1. The graph shows the temperature T°C of a metal bar t minutes after it leaves a furnace. It is modelled by \(T=d+Ae^{kt}\).

    510152020406080100tTT(0, 80)(5, 47)
    1. (a) Write down d and A.
    2. (b) Find k.
    3. (c) Find T when t = 12.

  2. A model is \(y=a+b\ln x\).

    1. (a) Show that doubling x adds \(b\ln2\) to y, whatever x is.
    2. (b) Find the increase in y when x doubles and b = 3.1.

jmaths.xyz · AI HL · 12.7.2 Exponential models, half-life and logarithmic models