SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Trigonometry, 2ADV T3 2022 HSC 23

The depth of water in a bay rises and falls with the tide. On a particular day the depth of the water, `d` metres, can be modelled by the equation

`d=1.3-0.6 cos((4pi)/(25)t)`,

where `t` is the time in hours since low tide.

  1. Find the depth of water at low tide and at high tide.  (2 marks)

  2. What is the time interval, in hours, between two successive low tides?  (1 mark)

  3. For how long between successive low tides will the depth of water be at least 1 metre?  (3 marks)

Show Answers Only
  1. `text{Low: 0.7 m,  High: 1.9 m}`
  2. `25/2\ text{hours}`
  3. `25/3\ text{hours}`
Show Worked Solution

a.   `text{S}text{ince}\ \ –1<=cos((4pi)/(25)t)<=1:`

`text{Low Tide}\ =1.3-0.6(1)=0.7\ text{m}`

`text{High Tide}\ =1.3-0.6(-1)=1.9\ text{m}`
 

b.   `text{Time between two low tides = Period of equation}\ (n)`

`(2pi)/n` `=(4pi)/25`  
`n/(2pi)` `=25/(4pi)`  
`n` `=25/2\ text{hours}`  

 


♦ Mean mark part (b) 47%.

c.   `text{Find}\ \ t\ \ text{when}\ \ d=1:`

`1.3-0.6 cos((4pi)/(25)t)` `=1`  
`-0.6 cos((4pi)/(25)t)` `=-0.3`  
`cos((4pi)/(25)t)` `=1/2`  

 

`(4pi)/(25)t` `=pi/3,\ \ (5pi)/3`  
`t` `=25/12,\ \ 125/12`  

 
`:.\ text{Time between low tides where water depth}\ >= 1\ text{m}`

`=125/12-25/12`

`=100/12`

`=25/3\ text{hours}`


♦ Mean mark part (c) 46%.

Trigonometry, 2ADV T3 SM-Bank 13

On any given day, the depth of water in a river is modelled by the function

`h(t) = 14 + 8sin((pit)/12),\ \ 0 <= t <= 24`

where `h` is the depth of water, in metres, and  `t`  is the time, in hours, after 6 am. 

  1. Find the minimum depth of the water in the river.  (1 mark)

  2. Find the values of  `t`  for which  `h(t) = 10`.  (2 marks)

Show Answers Only
  1. `6\ text(m)`
  2. `14quadtext(or)quad22`
Show Worked Solution

i.   `h_(text(min))\ text(occurs when)\ \ sin((pit)/12)=-1`

MARKER’S COMMENT: Students who used calculus to find the minimum were less successful.
`:. h_(text(min))` `= 14 – 8`
  `= 6\ text(m)`

 

ii.    `14 + 8sin(pi/12t)` `= 10`
  `sin(pi/12t)` `= – 1/2`

 

`text(Solve in general:)`

`pi/12t` `=(7pi)/6 + 2pi n\ \ \ \ text(or)\ \ \ `  `pi/12t` `= (11t)/6 + 2pi n,`
`t` `= 14 + 24n` `t` `=22 + 24n`

 

`text(Substitute integer values for)\ n,`

`:. t = 14quadtext(or)quad22,\ \ \ (0<=t<=24)`

Trigonometry, 2ADV T3 2009 HSC 7b

Between 5 am and 5 pm on 3 March 2009, the height, `h`, of the tide in a harbour was given by

`h = 1 + 0.7 sin(pi/6 t)\ \ \ text(for)\ \ 0 <= t <= 12`

where  `h`  is in metres and  `t`  is in hours, with  `t = 0`  at 5 am. 

  1. What is the period of the function  `h`?    (1 mark)

  2. What was the value of  `h`  at low tide, and at what time did low tide occur?     (2 marks)

  3. A ship is able to enter the harbour only if the height of the tide is at least 1.35 m.

     

    Find all times between 5 am and 5 pm on 3 March 2009 during which the ship was able to enter the harbour.    (3 marks)

Show Answers Only
  1. `12\ text(hours)`
  2. `text(2pm)\ \ text{(5am + 9 hours)}`
  3. `text(6am to 10am)`
Show Worked Solution

i.   `h = 1 + 0.7 sin (pi/6 t)\ \ text(for)\ 0 <= t <= 12`

`T` `= (2pi)/n\ \ text(where)\ n = pi/6`
  `= 2 pi xx 6/pi`
  `= 12\ text(hours)`

 

`:.\ text(The period of)\ h\ text(is 12 hours.)`

 

ii.  `text(Find)\ h\ text(at low tide)`

IMPORTANT: Using `sin x=–1` for a minimum here is very effective and time efficient. This property of trig functions is often very useful in harder questions.

`=> h\ text(will be a minimum when)`

`sin(pi/6 t) = -1`

`:.\ h_text(min)` `= 1 + 0.7(-1)`
  `= 0.3\ text(metres)`

 

`text(S)text(ince)\ \ sinx = -1\ \ text(when)\ \ x = (3pi)/2`

`pi/6 t` `= (3pi)/2`
`t` `= (3pi)/2 xx 6/pi`
  `= 9\ text(hours)`

 
`:.\ text{Low tide occurs at 2pm (5 am + 9 hours)}`


iii.  `text(Find)\ \ t\ \ text(when)\ \ h >= 1.35`

`1 + 0.7 sin (pi/6 t)` `>= 1.35`
`0.7 sin (pi/6 t)` `>= 0.35`
`sin (pi/6 t)` `>= 1/2`
`sin (pi/6 t)` `= 1/2\ text(when)`
`pi/6 t` `= pi/6,\ (5pi)/6,\ (13pi)/6,\ text(etc …)`
   
`t` `= 1,\ 5\ \ \ \ \ \ (0 <= t <= 12)`

 

Trig Calculus, 2UA 2009 HSC 7b Answer

`text(From the graph,)`

`sin(pi/6 t) >= 1/2\ \ \ text(when)\ \ 1 <= t <= 5`

 
`:.\ text(Ship can enter the harbour between 6 am and 10 am.)`