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

A bucket on a water wheel is at its lowest point, 0.5 m above the stream, at t = 0, where t is in seconds. Its highest point is 6.5 m above the stream. The wheel turns once every 12 seconds.

1. (a) Write down the model \(h=a\cos(bt)+d\).
2. (b) Find the height of the bucket at t = 4.

A wave passes a post. The water level h metres is highest at 7.4 and lowest at 2.6, and the time between crests is 5 seconds. At t = 0 the level is at its mean and rising.

1. (a) Write down the model \(h=a\sin(bt)+d\).
2. (b) Find the water level at t = 1.

A quantity is modelled by \(P=8\sin(30t)+10\), for one full cycle \(0\le t\le12\).

1. (a) Find the first time that P = 15.
2. (b) Find the total time in one cycle for which P is at least 15.

A Ferris wheel has a diameter of 40 m. Its lowest point is 3 m above the ground. It turns once every 6 minutes. A rider gets on at the lowest point at t = 0, where t is in minutes.

1. (a) Write down the model \(h=a\cos(bt)+d\).
2. (b) Find the height after 2.5 minutes.
3. (c) Find the time in one turn for which the height is more than 30 m.

The depth of water at a harbour entrance is 7.2 m at high tide and 2.4 m at low tide. The time between high tides is 12.4 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 = 2.
3. (c) A ship needs a depth of at least 5.5 m. Find the time in the first 12.4 hours for which it can enter.

The mean temperature T °C in a coastal town 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 2 °C and the highest is 22 °C.

1. (a) Write down the model \(T=a\cos(bt)+d\).
2. (b) Find the temperature when t = 3.5.
3. (c) Find the number of months in a year when the temperature is above 15 °C.
4. (d) Give one reason why the model may not be reasonable.

The depth D metres of water over a sandbank is modelled by \(D=2.4\sin(bt)+4.1\), where t is in hours and the time between high tides is 12.4 hours. A boat can cross when the depth is at least 3 m.

1. (a) Find the two times in the first cycle when the depth is 3 m.
2. (b) Find the total time in the first 24 hours for which the depth is at least 3 m.
3. (c) Say one way a real tide differs from this model.

A Ferris wheel has a radius of 15 m and its lowest point is 2 m above the ground. It turns once every 8 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.