Sinusoidal models: Ferris wheels, tides and daylight Problem set B

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 \(h=a\cos(bt)+d\).
2. (b) Find the height of the pedal at t = 0.5.

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 \(h=a\sin(bt)+d\).
2. (b) Find the height at t = 0.3.

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

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 \(h=a\cos(bt)+d\).
2. (b) Find the height after 6 seconds.
3. (c) Find the time in one turn for which the height is more than 7 m.

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 \(D=a\sin(bt)+d\).
2. (b) Find the depth at t = 3.5.
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.

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 \(T=a\cos(bt)+d\).
2. (b) Find the temperature when t = 2.5.
3. (c) Find the number of months in a year when the temperature is above 10 °C.
4. (d) Give one reason why the model may not be reasonable.

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.
2. (b) Find the total time in the first 24 hours for which the depth is at least 2.5 m.
3. (c) Say one way a real tide differs from this 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.
2. (b) Find the first time after t = 0 at which the two riders are at the same height.
3. (c) Find that height.