Question 1

In a certain city in the Northern hemisphere, the number of daylight hours H(t) at a time tdays of the year is given by

H(t) = A sin [ () t + c ] + D

wheret = 0 corresponds to January 1st. The maximum number of daylight hours occurs on June 21 st (or 171 days after January 1st) and is equal to 15 hours. The minimum number of daylight hours is equal to 11 hours.

  1. Find the values of A and D
  2. Find the smallest possible positive value of c, correct to two decimal places.
  3. Sketch the graph of H(t) for 0
  4. Find the percentage of a year in which the number of daylight hours in this city is greater than 12 hours.

Question 2

An old rock formation in an oil field has been warped by geological pressures into a sinusoidal shape. Over the centuries the top of the rock stratum has been eroded away, leaving the ground with a flat surface from which various rock formations are cropping out. Geologists have imposed an x-axis along the horizontal ground and a y axis at the edge of a rock outcropping and made measurments as indicated below. An oil well drilled at x = 100 reaches the top of the rock stratum underground at a vertical depth of 90 m.

  1. Determine the equation of the function which models the top of the rock stratum.
  2. How deep would a hill drilled at be if it reaches the top surface of the rock stratum? Give your answer to the nearest metre.
  3. i. What is the maximum depth of the top of the stratum?
  1. Assuming that the sinusoidal shape of the rock formation extends for , find the values of x for which the depth is at a maximum.
  1. How high above the present ground level did the stratum go before it was eroded away?
  2. For what values of x between 0 and 800 is the stratum within 120 m of the surface?
  3. A mineral exploration company has decided to drill for valuable minerals above the rock stratum. Vertical holes will be drilled at equally spaced intervals of 100 m along the x axis from x = 100 to x = 700 inclusive. The cost of drilling is $75 per metre. Calculate an estimate of the total cost of drilling.

Question 3

As a Ferris wheel rotates, the vertical height of a seat varies as a sine function. You are going for a ride on a Ferris wheel in the US (measurements in feet!). When the last seat is filled, and the Ferris wheel starts, you are in the position indicated in the diagram below:

Let tbe the number of seconds after the Ferris wheel starts. At t = 3, you first reach the top, which is 43 feet above the ground. The diameter of the wheel is 40 feet. The wheel makes a revolution every 24 seconds.

  1. Determine a function in the form which gives the distance d at time t.
  2. Sketch the graph of d versus t for one complete cycle.
  3. What is your height above the ground after 20 seconds? Give your answer in feet correct to two decimal places.
  4. Find the first three times at which your height is 18 feet. Give your answer in seconds correct to two decimal places.

Question 4

A tsunami is a high speed deep ocean wave caused by an earth quake. The water first goes down from its normal level then rises by the same amount above its normal level. The period of a tsunami is about 15 minutes. Suppose that at time t = 0 (minutes) a tsunami of amplitude 9.8 m strikes a beach where the water is 10m deep.

  1. What will be the maximum and minimum depths of water at the beach under the impact of the tsunami?
  2. Determine a function that will give the height of the water at time t minutes after the tsunami strikes.
  3. What are the first two times between which the level of the water at the beach will be less than 1 m? Give your answer in minutes correct to two decimal places.
  4. The wavelength of the tsunami is the distance between successive crests. If the tsunami is travelling at 1200 km/hr, what is its wavelength?

Question 5

Ona Nyland owns an island several hundred metres from the shore of a lake. The diagram below shows a cross section of the shore, lake and island. The island was formed millions of years ago by the stresses that caused the earth’s surface into a sinusoidal pattern shown. The highest point on the shore is at x = -150 metres. From measurements made on the near shore (solid line for the graph) topographers find that the equation of the sinusoidal is :

) where x and y are measured in metres.

  1. What is the highest level that the island rises above the water level in the lake? What is the x value at this point?
  2. What is the deepest the sinusoidal goes below the water level in the lake? What is the value of x at this point?
  3. Over the centuries, silt has filled the bottom of the lake so that water is only 40 m deep. That is, the silt line has equation: y =  40. If Ona goes scuba diving, between what values of x will he expect to find silt? Give your answers correct to the nearest metre.
  4. If Ona drills an offshore well at a distance of x = 700, through what depth of silt would he have to drill until he reaches the sinusoid? Give your answer to the nearest metre.
  5. The sinusoid appears to go through the origin. Does it actually do so, or does it just miss? Show your calculations.
  6. Find algebraically the interval of x values between which the island is at or above the water level. Give your answers correct to the nearest metre. State how wide the island is, from the water on one side to the water on the other side.

Question 6

For several hundred years, astronomers have kept track of the number of sunspots that occur on the surface of the sun. The number of sunspots in a given year varies periodically, from a minimum of about 10 a year to a maximum of about 110 a year.Between 1750 and 1948, there were exactly 18 complete cycles.

  1. What is the period of a sunspot cycle?
  2. Assume that the number of sunspots per year is sinusoidal function of time and that a maximum occurred in 1948. Find an equation of the form: which gives the number N of sunspots per year at year x (x = 0 is 0 AD)
  3. Use your equation to predict how many sunspots there wil be this year.
  4. How many sunspots are predicted for the year 2020?
  5. In how many years from now will there be about 53 sunspots?

Question 7

Naturalists are studying an Arctic fox population that appears to vary periodically with time. Let t = 0 the time (in years) at which records were first kept. A minimum number of 200 foxes appeared at t = 2.9 years. The next maximum, 800 foxes, appeared at t = 5.1 years,

a, Find an equation that gives the population P of foxes at time t years in the form:

, giving the values of b and k correct to one decimal place.

b. Predict the population of foxes after:

i. 7 years

ii. 8 years

iii. 9 years.

c. The population of foxes is declared vulnerable when the population drops below 300. Between what two non-negative values of t was the population first declared vulnerable? Give your answer correct to two decimal places.

Question 8

  1. A patient is undergoing chemotherapy treatment every 3 weeks. Initially, her red blood count has its maximum value of 900. The count drops to a low of 200 and then rises again to a maximum of 900 at t = 3 (weeks), when the next dosage is administered. Assuming that the red blood count R(t) is a sinusoidal function of time (measured in weeks), write down the function R(t).
  2. What is the red blood count after 7 weeks of treatment?
  3. The patient feels good if her red blood count is 750 or above. What is the longest continuous time interval during which she will feel good over the course of her treatment? Give your answer correct to four decimal places.
  4. What overall percentage of the time will she feel good? Give your answer correct to two decimal places.

Question 8

A hydro electric company is proposing to build a horizontal pipeline which will pass through a new tunnel and then over a bridge. The diagram below shows the cross section of proposed route with a tunnel through the mountain and a bridge over the valley to carry the pipe line.

The boundary of the cross section can be modelled by the function:

where

wherey is the height in metres above the proposed bridge and x is the distance in metres from a point O where the tunnel will start.

a, What is the height (in metres) of the top of the mountain above the bridge?

b. How many metres below the bridge is the bottom of the valley?

c. What is the exact length of

i. the tunnel

ii. the bridge

d. What would be the length, correct to the nearest metre, of the tunnel if it were built 20 metres higher up the mountain?

It is proposed to build the tunnel and bridge above the original proposed position:

  1. Suppose the tunnel is built instead at a height on the mountain such that it starts where x = k , .
  1. Find the length of the tunnel in terms of k.
  2. Find the length of the bridge in terms of k.

f. The estimated total cost, C thousand dollars, for building the tunnel and bridge for this second proposal is equal to the sum of the square of the length (in metres) of the tunnel and the square of the length (in metres) of the bridge. Write down an expression for the estimated total cost of building the tunnel and the bridge if the tunnel starts when x = k, in terms of k.

g. Hence find the exact value of k for which the estimated cost of the proposal will be a minimum.

Graphing Factorizable Trig equations

SINE RULE APPLCIATION PROBLE<S

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