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Algebra, STD2 EQ-Bank 30

The number \((F)\) of fish in Lake Mulloway is decreasing. The number of fish is modelled using

\(F=k(1.05)^{-t}\)  for  \( t \geq 0,\)

where \(t\) is time in years.

The number of fish in Lake Mulloway after 1 year is 4620.

  1. Using the model, estimate the number of fish in Lake Mulloway today.   (2 marks)

  2. The local newsgroup on socials has used the model to predict the number of fish that will be in Lake Mulloway in 40 years time.
  3. If the number of fish in the lake needs to always be above 1000, explain why this model is NOT appropriate to use in this case.   (1 mark)
Show Answers Only

a.    \(F=4851\)

b.    \(\text{When}\ \ t=40\ \ \Rightarrow\ \ F \approx 689\)

\(\text{Since the number of fish needs to always be over 1000, it is not appropriate.}\)

Show Worked Solution

a.    \(\text{When}\ \ t=1,\ \ F=4620:\)

\(4620=k(1.05)^{-1}\ \ \Rightarrow\ \ k=4620 \times 1.05=4851\)

\(\text{Find}\ F\ \text{when}\ \ t=0:\)

\(F=4851(1.05)^{0}=4851\)
 

b.    \(\text{When}\ \ t=40:\)

\(F=4851(1.05)^{-40} \approx 689\)

\(\text{Since the number of fish needs to always be over 1000, it is not appropriate.}\)

Algebra, STD2 EQ-Bank 29

The population ( \(P\) ) of a town is reducing. The population is modelled using

\(P=k(1.1)^{-t}\)  for  \( t \geq 0,\)

where \(t\) is time in years.

The population of the town today is 5000 .

  1. Use the model to estimate the population of the town 15 years from today.   (2 marks)

  2. The population of the town needs to always be above 500 .
  3. Explain why this model is NOT appropriate to use in this case.   (1 mark)

Show Answers Only

a.    \(P=5000(1.1)^{-15}=1196.96…=1197\ \text{(nearest person)}\)

b.    \(\text{When}\ \ t=30\ \ \Rightarrow\ \ P \approx 287\)

\(\text{Since the population needs to always be over 500, it is not appropriate.}\)

Show Worked Solution

a.    \(\text{When}\ \ t=0,\ \ P=5000:\)

\(5000=k(1.1)^{0}\ \ \Rightarrow\ \ k=5000\)

\(\text{Find}\ P\ \text{when}\ \ t=15:\)

\(P=5000(1.1)^{-15}=1196.96…=1197\ \text{(nearest person)}\)
 

b.    \(\text{When}\ \ t=30\ \ \Rightarrow\ \ P \approx 287\)

\(\text{Since the population needs to always be over 500, it is not appropriate.}\)