Mathematical Methods Unit 2H9- Circular Functions modelling questions

Example 1The diagram shows the heart rate of an athlete during a particular hour of a workout.

(a) Find the initial heart rate.(b) State i the amplitude and ii the period.(c) Express H as a function of t.

Example 2 A guitar is vibrating so that the position (in mm) of the string at any time t seconds is given by the function p(t) = 7sin 880t.

(a) What is its period?(b) How many times will it go through its cycle each second?(c) What will be the position of the string at time t = 0.02 seconds correct to two decimal places?(d) When is its position at zero for the second time?

Example 3 The current passing through a capacitor is given by I = 4sin(100t) where I amperes is the current t seconds after the capacitor is activated.

(a) Find the current after

11000

second correct to 4 decimal places.

(b) After what time does the current first change direction (ie pass from positive to negative)?(c) How much time elapses between each change of current direction?(d) How many times does the current reach its maximum value in one second?

Example 4 A student wanting to catch fish to sell at a local market on Sunday has discovered that more fish are in the water at the end of the pier when the depth of water is greater than 8.5 metres. The depth of the water

(in metres) is given by d=7+3 sin

π6t, where t hours is the number of hours after midnight on Friday.

(a) What is the maximum and minimum depth of the water at the end of the pier?(b) Sketch a graph of d against t from midnight on Friday until midday on Sunday.(c) When does the water first reach maximum depth?(d) Between what hours should the student be on the pier in order to catch the most fish?(e) If the student can fish for only two hours at a time, when should she fish in order to sell the freshest

fish at the market from 1.00 am on Sunday morning?

Example 5A rock wall on the shore at Port Melbourne beach is submerged for part of the day. The height, h, of the water above the wall, in metres, can be approximated by the formula.

h=15 sinπ6t where t is the number of hours after 6:00 am on 1st January.

(a) Determine the amplitude and period of the graph.(b) Draw a graph of h for 0 ¿ t ¿ 24.(c) Find the time of the high tide and the approximate height of the water above the wall at high tide on

that day.(d) A boat can only dock at the shore when the tide is at least 10 metres above the rock wall. Find

when the boat can dock on that day.

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