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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, metres, can be modelled by the equation

,

where 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)

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a.   


 

b.   

 
 
 

 


♦ Mean mark part (b) 47%.

c.   

 
 
 

 

 
 

 


♦ 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

where is the depth of water, in metres, and    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    for which  .  (2 marks)

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i.   

MARKER’S COMMENT: Students who used calculus to find the minimum were less successful.
 

 

ii.   
 

 

 

 

Trigonometry, 2ADV T3 2009 HSC 7b

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

where    is in metres and    is in hours, with    at 5 am. 

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

  2. What was the value of    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)

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i.   

 
 

 

 

ii. 

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

 

 

 

 


iii. 

   

 

Trig Calculus, 2UA 2009 HSC 7b Answer