Sinusoidal and logistic models Problem set B

A quantity P is modelled by \(P=a\sin(b(t-c))+d\). The amplitude is 7, the central line is P = 25, the period is 20 and the phase shift is 3.

1. (a) Write down the model.
2. (b) Find P when t = 8.

A quantity P is modelled by \(P=a\sin(b(t-c))+d\). Its maximum is 48 and its minimum is 12. The period is 8 and the phase shift is 1.

1. (a) Write down the model.
2. (b) Find P when t = 2.5.

A quantity P is modelled by \(P=a\sin(b(t-c))+d\). The mean is 55, the amplitude is 15 and the period is 10. At t = 2 the quantity is at its mean and rising.

1. (a) Write down the model.
2. (b) Find P when t = 3.5.

A population is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\), with t in years. The carrying capacity is 350. The population is 14 at t = 0 and 60 at t = 3.

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

A population is modelled by \(P=\dfrac{640}{1+9e^{-0.8t}}\), where t is in hours.

1. (a) Find P when t = 3.
2. (b) Find t when P = 400.

The pedal of a bicycle has a mean height of 0.45 m above the ground and an amplitude of 0.17 m. One turn takes 0.8 seconds. At t = 0.1 seconds the pedal is at its mean height and rising. Model the height as \(h=a\sin(b(t-c))+d\).

1. (a) Write a model for the height h m of the pedal, t seconds after the start.
2. (b) Find the height at t = 0.35.

The electricity demand in a city has a maximum of 52 GW and a minimum of 28 GW. The period is 24 hours and the phase shift is 6 hours. Model it as \(E=a\sin(b(t-c))+d\).

1. (a) Write a model for the demand \(E\) GW, t hours after midnight.
2. (b) Find the demand at t = 10.

The number of fish in a lake is modelled by \(P=\dfrac{L}{1+Ce^{-kt}}\), with t in months. The lake can support 1200 fish. There are 30 at t = 0 and 100 at t = 3.

1. (a) Find C and k.
2. (b) Predict the population at t = 6.

A mass hangs on a spring and bobs up and down. Its amplitude is 0.12 m, its mean height is 1.5 m, the period is 0.9 s and the phase shift is 0.1 s. Model the height as \(h=a\sin(b(x-c))+d\).

1. (a) Write a model for the height h m of the mass at time x s.
2. (b) Find h when x = 0.4.

The graph shows a logistic model \(P=\dfrac{400}{1+7e^{-0.8t}}\) (solid) and an exponential model with the same start (dashed), for a population of frogs.

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.

A Ferris wheel carriage has height \(h=9\sin(b(t-2))+10\) metres, with period 20 seconds, where t is in seconds.

1. (a) Find the first time after t = 0 when the carriage is 14 m above the ground.
2. (b) Find the second time in the same turn when the carriage is 14 m above the ground.