Sinusoidal and logistic models Problem set A One question at a time

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.

   • a = 4, d = 10, c = 2, \(b=2\pi\div15=0.419\)
   • \(P=4\sin(0.419(t-2))+10\)

2. (b) Find P when t = 5.

   • \(P=4\sin(0.419\times3)+10=13.8\)

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.

   • \(a=(31-9)\div2=11\), \(d=(31+9)\div2=20\), \(b=2\pi\div12=0.524\)

2. (b) Find P when t = 7.

   • \(P=11\sin(0.524(7-3))+20=29.5\)

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.

   • d = 20, a = 6, \(b=2\pi\div8=0.785\)
   • The curve is on the mean and rising at t = 1.5, so c = 1.5
   • \(P=6\sin(0.785(t-1.5))+20\)

2. (b) Find P when t = 4.

   • \(P=6\sin(0.785\times2.5)+20=25.5\)

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.

   • At t = 0: \(25=\dfrac{500}{1+C}\), so C = 500 ÷ 25 − 1 = 19

2. (b) Find k.

   • \(80=\dfrac{500}{1+19e^{-2k}}\)
   • Solve for k: k = 0.643

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

1. (a) Find P when t = 6.

   • \(P=\dfrac{900}{1+14e^{-0.45\times6}}=464\)

2. (b) Find t when P = 600.

   • \(\dfrac{900}{1+14e^{-0.45t}}=600\)
   • Solve for t: t = 7.40

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.

   • d = 3.2, a = 1.4, \(b=2\pi\div12.4=0.507\), c = 1.1
   • \(D=1.4\sin(0.507(t-1.1))+3.2\)

2. (b) Find the depth at t = 5.

   • \(D=1.4\sin(0.507\times3.9)+3.2=4.49\)

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.

   • \(a=(38-14)\div2=12\), \(d=(38+14)\div2=26\), \(b=2\pi\div24=0.262\)
   • \(T=12\sin(0.262(t-9))+26\)

2. (b) Find the temperature at t = 12.

   • \(T=12\sin(0.262\times3)+26=34.5\)

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.

   • \(C=2000\div50-1=39\)
   • \(190=\dfrac{2000}{1+39e^{-4k}}\): k = 0.352

2. (b) Predict the population at t = 8.

   • \(P=\dfrac{2000}{1+39e^{-0.352\times8}}=601\)

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.

   • a = 0.35, d = 1.2, \(b=2\pi\div3=2.09\), c = 0.5
   • \(h=0.35\sin(2.09(x-0.5))+1.2\)

2. (b) Find h when x = 1.7.

   • \(h=0.35\sin(2.09\times1.2)+1.2=1.41\)

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.

   • The logistic curve levels off at the dashed line: L = 600

2. (b) Find the initial population.

   • \(P(0)=\dfrac{600}{1+19}=30\)

3. (c) Find when the logistic population grows fastest, and the population then.

   • Fastest growth at \(P=L\div2=300\)
   • \(t=\ln19\div0.5=5.89\)

4. (d) Say how the two models differ for large t.

   • The exponential model has no limit. The logistic model levels off at 600.

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.

   • \(1.4\sin(0.507(t-1.1))+3.2=4\)
   • Solve on the calculator, or graph both sides: first crossing t = 2.30

2. (b) Find the second time in the same cycle when the depth is 4 m.

   • The curve falls back to the same value at the second crossing: t = 6.10