2. An oceanographer is out at sea on a boat to take some temperature readings. A thermometer and a weight is attached to

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2 An Oceanographer Is Out At Sea On A Boat To Take Some Temperature Readings A Thermometer And A Weight Is Attached To 1 (60.6 KiB) Viewed 54 times

2. An oceanographer is out at sea on a boat to take some temperature readings. A thermometer and a weight is attached to a cable, and they are lowered into the water by a winch on the boat. In order to receive meaningful data, the thermometer must be positioned between 8 and 12 metres above the ocean floor. However, the ocean's tides cause the boat to rise and lower itself relative to the ocean floor, and hence the thermometer moves outside of the desired depth range. In order to counteract this effect, the winch is set to extend and retract the cable in oscillation. The depth of the water is 50 metres plus y1​(t), a cosine function with an amplitude of 4 metres, a period of 12 hours, and a phase time of 1 hour. The length of cable that is submerged underwater is 40 metres plus y2​(t), a cosine function with an aroplitude of 5 metres, a period of 12 hours, and a phase time of 0 hours. The distance from the ocean floor to the thermometer is 10 metres plus y3​(t)=y1​(t)−y2​(t) (a) Determine the expressions for y1​(t) and y2​(t) (in meters) at any time t (in hours). (b) Determine an expression for y3​(t) of the form y3​(t)=Asin(ωt+α)1​ where A>0,ω>0, and α∈(−π,π]. (c) Does the thermometer remain between the desired depths? Justify your response.