Sinusoidal and logistic models Problem set A

A quantity P is modelled by \(P=a\sin(b(t-c))+d\). The amplitude is 4, the central line is P = 10, the period is 15 and the phase shift is 2.

1. (a) Write down the model.
2. (b) Find P when t = 5.

A quantity P is modelled by \(P=a\sin(b(t-c))+d\). Its maximum is 31 and its minimum is 9. The period is 12 and the phase shift is 3.

1. (a) Write down the model.
2. (b) Find P when t = 7.

A quantity P is modelled by \(P=a\sin(b(t-c))+d\). The mean is 20, the amplitude is 6 and the period is 8. At t = 1.5 the quantity is at its mean and rising.

1. (a) Write down the model.
2. (b) Find P when t = 4.

A population is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\), with t in years. The carrying capacity is 500. The population is 25 at t = 0 and 80 at t = 2.

1. (a) Find C.
2. (b) Find k.

A population is modelled by \(P=\dfrac{900}{1+14e^{-0.45t}}\), where t is in hours.

1. (a) Find P when t = 6.
2. (b) Find t when P = 600.

The depth of water at a jetty has a mean of 3.2 m and an amplitude of 1.4 m. The time between high tides is 12.4 hours. At t = 1.1 hours the depth equals the mean and is rising. Model the depth as \(D=a\sin(b(t-c))+d\).

1. (a) Write a model for the depth D m, t hours after midnight.
2. (b) Find the depth at t = 5.

The temperature in a desert has a maximum of 38°C and a minimum of 14°C. The period is 24 hours and the phase shift is 9 hours. Model it as \(T=a\sin(b(t-c))+d\).

1. (a) Write a model for the temperature \(T\)°C, t hours after midnight.
2. (b) Find the temperature at t = 12.

A colony of bacteria in a dish is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\), with t in hours. The dish can hold 2000 bacteria. There are 50 at t = 0 and 190 at t = 4.

1. (a) Find C and k.
2. (b) Predict the population at t = 8.

A wave travels along a rope. The amplitude is 0.35 m, the rope's rest height is 1.2 m, the wavelength is 3 m and the phase shift is 0.5 m. Model the height as \(h=a\sin(b(x-c))+d\).

1. (a) Write a model for the height h m of the rope at distance x m.
2. (b) Find h when x = 1.7.

The graph shows a logistic model \(P=\dfrac{600}{1+19e^{-0.5t}}\) (solid) and an exponential model with the same start (dashed), for a population of insects.

1. (a) Write down the carrying capacity.
2. (b) Find the initial population.
3. (c) Find when the logistic population grows fastest, and the population then.
4. (d) Say how the two models differ for large t.

At a jetty the depth is \(D=1.4\sin(b(t-1.1))+3.2\), with period 12.4 hours. Boats can dock when the depth is at least 4 m.

1. (a) Find the first time after t = 0 when the depth is 4 m.
2. (b) Find the second time in the same cycle when the depth is 4 m.