Topic 2.9: HL modelling (HL only) Questions

In a logistic model y = L/(1 + C·e^−kt), the value of y as t → ∞ is:

A town's population is modelled by P(t) = 12·$e^{0.03t}$ thousand, where t is years after 2020. State the population in 2020 and the continuous growth rate, and interpret the rate.

A quantity DOUBLES every fixed time period. The best model is:

For a sinusoidal model a·sin(b(t − c)) + d, the period of the cycle is:

Tidal depth at a harbour is modelled by D(t) = 4·sin(0.5t) + 9 metres, with t in hours. State the amplitude, the midline, the maximum depth, and the period.

A phone plan charges \$0.20 per minute for the first 100 minutes, then \$0.10 per minute after that. Write a piecewise cost function C(t) in dollars for t minutes, and find the cost of 250 minutes.

The spread of a rumour in a school of 1200 students is modelled by N(t) = 1200/(1 + 80·e^−0.9t), where t is in days. Find the number who know the rumour after 3 days, and state the carrying capacity.

A graph of data rises, falls, then rises again (two turning points). The simplest polynomial model is:

A bacteria count is modelled by N = k·$e^{rt}$ (N in thousands, t in hours). At t = 0, N = 5; at t = 4, N = 40. Find k and r, then interpret r.

A company fits an exponential model to 5 years of revenue data and uses it to predict revenue 40 years ahead. The best comment on this is:

Find $a$ and $b$.

Solar panel output over a day is modelled by P(t) = 6·sin(0.26(t − 6)) + 1 (kW), valid for daylight hours 6 ≤ t ≤ 18 (t = hours after midnight). Find the time of peak output and comment on a limitation of using this model at night.

Find $k$ and $a$, the concentration at $t=0$, and the time at which the concentration falls to $5$ mg/L.