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Trigonometry, 2ADV T3 2020 HSC 31

The population of mice on an isolated island can be modelled by the function.

`m(t) = a sin (pi/26 t) + b`,

where  `t`  is the time in weeks and  `0 <= t <= 52`. The population of mice reaches a maximum of 35 000 when  `t=13`  and a minimum of 5000 when  `t = 39`. The graph of  `m(t)`  is shown.
 

  1. What are the values of `a` and `b`?  (2 marks)

  2. On the same island, the population of cats can be modelled by the function
     
    `\ \ \ \ \ c(t) = −80cos(pi/26 (t - 10)) + 120`
     
    Consider the graph of  `m(t)`  and the graph of  `c(t)`.

     

    Find the values of  `t, \ 0 <= t <= 52`, for which both populations are increasing.  (3 marks)

  3. Find the rate of change of the mice population when the cat population reaches a maximum.  (2 marks)

Show Answers Only
  1. `a = 15\ 000, b = 20\ 000`
  2. `text(Both populations are increasing when)\ 10 < t < 13`
  3. `\text(– 643 mice per week)`
Show Worked Solution
a.    `b` `= (35\ 000 + 5000)/2`
    `= 20\ 000`

 

`a` `=\ text(amplitude of sin graph)`
  `= 35\ 000 – 20\ 000`
  `= 15\ 000`

 

b.   `text(By inspection of the)\ \ m(t)\ \ text(graph)`

♦♦ Mean mark part (b) 30%.

`m^{′}(t) > 0\ \ text(when)\ \ 0 <= t < 13\ \ text(and)\ \ 39 < t <= 52`

`text(Sketch)\ \ c(t):`

`text(Minimum)\ \ (cos0)\ \ text(when)\ \ t = 10`

`text(Maximum)\ \ (cospi)\ \ text(when)\ \ t = 36`

`:. c^{′}(t) > 0\ \ text(when)\ \ 10 < t < 36`

`:. text(Both populations are increasing when)\ \ 10 < t < 13`

 

c.   `c(t)\ text(maximum when)\ \ t = 36`

♦♦♦ Mean mark part (c) 27%.
`m(t)` `= 15\ 000 sin(pi/26 t) + 20\ 000`
`m^{′}(t)` `= (15\ 000pi)/26 cos(pi/26 t)`
`m^{′}(36)` `= (15\ 000pi)/26 · cos((36pi)/26)`
  `= -642.7`

 
`:.\ text(Mice population is decreasing at 643 mice per week.)`

Calculus, 2ADV C3 2004 HSC 5b

A particle moves along a straight line so that its displacement, `x` metres, from a fixed point `O` is given by  `x = 1 + 3 cos 2t`, where  `t`  is measured in seconds.

  1. What is the initial displacement of the particle?  (1 mark)

  2. Sketch the graph of  `x`  as a function of  `t`  for  `0 ≤ t ≤ pi`.  (2 marks)

  3. Hence, or otherwise, find when AND where the particle first comes to rest after  `t = 0`.  (2 marks)

  4. Find a time when the particle reaches its greatest magnitude of velocity. What is this velocity?  (2 marks)

Show Answers Only
  1. `text(4 m to the right of)\ O.`
  2. `text(See Worked Solutions)`
  3. `t = pi/2\ text(seconds, 2 m to the left of)\ O.`
  4. `text(6 m s)^(−1)`
Show Worked Solution

i.   `x = 1 + 3 cos 2t`

`text(When)\ \ t = 0,`

`x` `= 1 + 3 cos 0`
  `= 1 + 3`
  `= 4`

 

`:.\ text(Initial displacement is 4 m to the right of)\ O.`

 

 ii.  `text(Period)\ = (2pi)/n = (2pi)/2 = pi`

`text(Considering the range)`

`-1` `<=cos 2t<=1`
`-3` `<=3cos 2t<=3`
`-2` `<=1 + 3 cos 2t<=4`

 

 Calculus in the Physical World, 2UA 2004 HSC 5b

 

iii.   `x` `= 1 + 3 cos 2t`
  `:.v` `= −6 sin 2t`

 

`text(The particle comes to rest when)\ \ v=0`

`-6 sin 2t` `= 0`
`sin 2t` `= 0`
`2t` `= 0, pi, 2pi…`
`t` `= 0, pi/2, pi…`

 

`:.\ text(After)\ \ t=0, text(particle first comes to rest when)`

`t = pi/2\ text(seconds.)`

`text(When)\ t = pi/2,`

`x` `= 1 + 3 cos 2(pi/2)`
  `= 1 + 3 cos pi`
  `= 1 + 3(−1)`
  `= −2`

 

`:.\ text(Particle first comes to rest at 2 m to the left of)\ O.`

 

iv.  `x = 1 + 3 cos 2t`

`v` `= -6 sin 2t`
`a` `= -12 cos 2t`
   

`text(MAX occurs when)\ \ a=0`

`−12 cos 2t` `= 0`
`cos 2t` `= 0`
`2t` `= pi/2, (3pi)/2, …`
`t` `= pi/4, (3pi)/4, …`

 
`:.\ text(Maximum at)\ \ t=pi/4,\ \ (3pi)/4, …\ text(seconds,)`

 

`text(When)\ \ t = pi/4\ text(seconds,)`

`v` `= -6 sin 2(pi/4)`
  `= -6 sin(pi/2)`
  `= −6`

 
`:.\ text(Maximum is 6 m s)^(−1).`

Calculus, 2ADV C4 2015 HSC 15c

Water is flowing in and out of a rock pool. The volume of water in the pool at time `t` hours is `V` litres. The rate of change of the volume is given by

`(dV)/(dt) = 80 sin(0.5t)`

At time  `t = 0`, the volume of water in the pool is 1200 litres and is increasing.

  1. After what time does the volume of water first start to decrease?  (2 marks)

  2. Find the volume of water in the pool when  `t = 3`.  (2 marks)

  3. What is the greatest volume of water in the pool?  (1 mark)

Show Answers Only
  1. `2 pi\ text(hours)`
  2. `1349\ text(litres)`
  3. `1520\ text(litres)`
Show Worked Solution

i.   `(dV)/(dt) = 80 sin (0.5t)`

♦ Mean mark (i) 37%.

`text(Volume decreases when)\ \ (dV)/(dt) < 0`

`(dV)/(dt) < 0\ \ text(when)\ \ t > 2 pi\ text(hours)`

`:.\ text(After 2)\pi\ \text(hours, volume starts to decrease.)`

 

ii.   `V` `= int (dV)/(dt)\ dt`
  `= int 80 sin (0.5t)\ dt`
  `= -160 cos (0.5t) + c`

 

`text(When)\ \ t = 0,\ V = 1200`

`1200` `= -160 cos 0 + c`
`c` `= 1360`
`:.\ V` `= -160 cos (0.5t) + 1360`

 

`text(When)\ \ t = 3`

`V` `= -160 cos (0.5 xx 3) + 1360`
  `= 1348.68…\ \ text(litres)`
  `= 1349\ text{litres  (nearest litre)}`

 

iii.  `V = -160 cos (0.5t) + 1360`

♦♦ Mean mark (iii) 27%.

`=>\ text(Greatest volume occurs when)`

`cos (0.5t) = -1`

`:.\ text(Maximum volume)`

`= -160 (-1) + 1360`

`= 1520\ text(litres)`