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Calculus, 2ADV C4 2023 HSC 13

Let `P(t)` be a function such that `(dP)/(dt)=3000 e^{2t}`.

When `t=0, P=4000`.

Find an expression for `P(t)`.  (2 marks)

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`P(t)=1500e^{2t}+2500`

Show Worked Solution
`P(t)` `=int (dP)/(dt)\ dt`  
  `=int 3000e^{2t}\ dt`  
  `=1500e^{2t}+c`  

 
`text{When}\ t=0, P=4000`

`4000` `=1500e^0+c`  
`c` `=2500`  

 
`:.P(t)=1500e^{2t}+2500`

Calculus, 2ADV C4 EQ-Bank 2

The population, `D`, of Tasmanian Devils in a sanctuary is given by  `D(t)`, where  `t`  is the time in years after the sanctuary was established.

The devil population changes at a rate modelled by the function  `(dD)/(dt) = 28 e^(0.35t)`.

Calculate the increase in the number of Tasmanian Devils at the end of the first 8 years. Give your answer correct to three significant figures.  (3 marks)

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`1240 \ text((to 3 sig. fig.))`

Show Worked Solution
`int_0^8 28e^(0.35t)` `= [28 xx (1)/(0.35) e^(0.35t)]_0^8`
  `= 80(e^(0.35 xx 8) – e°)`
  `= 80(16.44 … – 1)`
  `= 1235.57 …`
  `= 1240 \ text((to 3 sig. fig.))`