Sinusoidal models: Ferris wheels, tides and daylight Problem set B One question at a time

A pedal of a bicycle is at its lowest point, 0.12 m above the ground, at t = 0, where t is in seconds. Its highest point is 0.52 m above the ground. The pedal goes round once every 1.5 seconds.

1. (a) Write down the model \(h=a\cos(bt)+d\).

   • a = (0.52 − 0.12) ÷ 2 = 0.2, d = (0.52 + 0.12) ÷ 2 = 0.32, b = 360 ÷ 1.5 = 240
   • It starts at the minimum, so a is negative.

2. (b) Find the height of the pedal at t = 0.5.

   • h = 0.32 − 0.2cos(240 × 0.5) = 0.42

A mass on a spring moves up and down. Its height h metres is highest at 1.2 and lowest at 0.4, and one bounce takes 2 seconds. At t = 0 the mass is at its middle height and moving up.

1. (a) Write down the model \(h=a\sin(bt)+d\).

   • a = (1.2 − 0.4) ÷ 2 = 0.4, d = (1.2 + 0.4) ÷ 2 = 0.8, b = 360 ÷ 2 = 180
   • It starts at the mean and rising, so sine.

2. (b) Find the height at t = 0.3.

   • h = 0.4sin(180 × 0.3) + 0.8 = 1.12

A quantity is modelled by \(P=6\sin(45t)+9\), for one full cycle \(0\le t\le8\).

1. (a) Find the first time that P = 12.5.

   • Solve for t: 6sin(45t) + 9 = 12.5: t = 0.793 (the first solution)

2. (b) Find the total time in one cycle for which P is at least 12.5.

   • The other solution in the cycle is t = 3.207
   • Time above = 3.207 − 0.793 = 2.41

A water wheel at a museum has a diameter of 10 m. Its lowest point is 0.8 m above the water. It turns once every 20 seconds. A paddle is at the lowest point at t = 0, where t is in seconds.

1. (a) Write down the model \(h=a\cos(bt)+d\).

   • a = (10.8 − 0.8) ÷ 2 = 5, d = 5.8, b = 360 ÷ 20 = 18
   • Starts at the lowest point: a is negative.

2. (b) Find the height after 6 seconds.

   • h = 5.8 − 5cos(18 × 6) = 7.35

3. (c) Find the time in one turn for which the height is more than 7 m.

   • Solve for t: h = 7 gives t = 5.77 and t = 14.23
   • Time above = 8.46

The depth of water at a river mouth is 4.6 m at high tide and 1.2 m at low tide. The time between high tides is 12 hours. At t = 0, where t is in hours, the depth is at its mean and rising.

1. (a) Write down the model \(D=a\sin(bt)+d\).

   • a = 1.7, d = 2.9, b = 360 ÷ 12 = 30

2. (b) Find the depth at t = 3.5.

   • D = 1.7sin(30 × 3.5) + 2.9 = 4.54

3. (c) A barge needs a depth of at least 3.5 m. Find the time in the first 12 hours for which it can pass.

   • Solve for t: D = 3.5 gives t = 0.689 and t = 5.31
   • 5.31 − 0.689 = 4.62

The mean temperature T °C in a mountain village t months after the middle of January, when it is at its lowest, is modelled with a period of 12 months. The lowest mean temperature is −6 °C and the highest is 18 °C.

1. (a) Write down the model \(T=a\cos(bt)+d\).

   • a = 12, d = 6, b = 360 ÷ 12 = 30
   • Starts at the minimum: −a cosine.

2. (b) Find the temperature when t = 2.5.

   • T = 6 − 12cos(30 × 2.5) = 2.89

3. (c) Find the number of months in a year when the temperature is above 10 °C.

   • Solve for t: T = 10 gives t = 3.65 and t = 8.35
   • Difference = 4.70

4. (d) Give one reason why the model may not be reasonable.

   • The model assumes every year is the same.

The depth D metres of water over a reef is modelled by \(D=1.8\sin(bt)+3\), where t is in hours and the time between high tides is 12 hours. A diving boat can pass when the depth is at least 2.5 m.

1. (a) Find the two times in the first cycle when the depth is 2.5 m.

   • Solve for t: D = 2.5 gives t = 6.54 and t = 11.5

2. (b) Find the total time in the first 24 hours for which the depth is at least 2.5 m.

   • Use the graph on 0 ≤ t ≤ 24 and find every crossing: 6.54, 11.5, 18.5, 23.5
   • Add the times above 2.5: 14.2 hours

3. (c) Say one way a real tide differs from this model.

   • Every cycle is the same in the model.

A big wheel has a radius of 9 m and its lowest point is 1 m above the ground. It turns once every 12 minutes. Rider 1 gets on at the lowest point at t = 0. Rider 2 is at the side of the wheel, level with the centre and going up, at t = 0.

1. (a) Write down a model for each rider.

   • Rider 1 starts at the lowest point: −9cos. Rider 2 starts at the mean going up: 9sin
   • d = 10, b = 360 ÷ 12 = 30

2. (b) Find the first time after t = 0 at which the two riders are at the same height.

   • Intersection of the two graphs: t = 4.5

3. (c) Find that height.

   • h = 10 − 9cos(30 × 4.5) = 16.4