Question

1. a seat’s position on a ferris wheel can be modelled by the function ( h(t) = 14 sin(2.1t + 0.8) + 15.5 ) where ( h(t) ) represents the height in metres and ( t ) represents the time in minutes. what is the height of the seat, to the nearest tenth of a metre, after 1.5 min? (2 marks)
2. captain picard is sitting in an inner tube in the wave pool. the depth of the water below him, in terms of time, during a series of waves can be represented by the graph shown.

a) what is the depth of the water below picard when no waves are being generated? explain how you know. (2 marks)
b) how high is each wave? show your work. (2 marks)

Explanation:

Step1: Substitute $t=1.5$ into $h(t)$

$h(1.5) = 14\sin(2.1\times1.5 + 0.8) + 15.5$

Step2: Calculate the argument of sine

$2.1\times1.5 + 0.8 = 3.15 + 0.8 = 3.95$

Step3: Compute sine of the value

$\sin(3.95) \approx -0.7055$

Step4: Calculate the height

$h(1.5) \approx 14\times(-0.7055) + 15.5 = -9.877 + 15.5 = 5.623$

Step5: Round to nearest tenth

$h(1.5) \approx 5.6$

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a) The no-wave depth is the midline of the sinusoidal graph, calculated as the average of the maximum and minimum depths. The max depth is 2.8 m, min is 1.2 m; their average is the equilibrium depth.
b) Wave height is the difference between the maximum depth and the equilibrium (no-wave) depth, or half the total distance between max and min depths.

Answer:

1. $5.6$ metres

1. a) 2.0 metres. This is the midline of the sinusoidal graph, found by $\frac{2.8 + 1.2}{2} = 2.0$, which is the stable depth without waves.

b) 0.8 metres. Calculation: $2.8 - 2.0 = 0.8$ (or $\frac{2.8 - 1.2}{2} = 0.8$)