Further models: natural log, logistic, piecewise, half-life practice questions.

A radioactive isotope has a half-life of 8 days. An initial sample of 240 g decays according to A(t) = 240 × (1/2)^(t/8). Calculate the mass remaining after 24 days.

• 30 g
• 60 g
• 80 g
• 120 g

Answer: 30 g

24 days is 3 half-lives (24/8 = 3), so the mass halves three times: 240 → 120 → 60 → 30 g.

A logistic model P(t) = L / (1 + Ce^(−kt)) describes population growth. State what the parameter L represents.

• The initial population
• The carrying capacity
• The growth rate
• The time to reach half the maximum

Answer: The carrying capacity

As t → ∞, the exponential term vanishes and P(t) → L, so L is the carrying capacity or maximum sustainable population.

The temperature of a cooling drink is modelled by T(t) = 20 + 60e^(−0.05t), where t is in minutes. State the temperature the drink approaches as t becomes very large.

• 0 °C
• 20 °C
• 60 °C
• 80 °C

Answer: 20 °C

As t → ∞, e^(−0.05t) → 0, so T → 20 °C, the ambient (room) temperature.

A quantity grows so that its natural-log model is y = 3 ln(x) + 5. Calculate y when x = e² (give an exact value).

• 11
• 8
• 13
• 6

Answer: 11

ln(e²) = 2, so y = 3(2) + 5 = 11.

A piecewise function is defined by f(x) = 2x + 1 for x < 3 and f(x) = x² − 2 for x ≥ 3. Calculate f(3).

• 7
• 6
• 5
• 9

Answer: 7

Since x = 3 satisfies x ≥ 3, use the second piece: f(3) = 3² − 2 = 9 − 2 = 7.

A sample of a substance decays with half-life 5 years. Determine the fraction of the original amount remaining after 15 years.

• 1/8
• 1/16
• 1/6
• 1/4

Answer: 1/8

15/5 = 3 half-lives, so the remaining fraction is (1/2)³ = 1/8.