The graph shows the populations of two different animals, \(W\) and \(K\), in a conservation park over time. The \(y\)-axis is the size of the population and the \(x\)-axis is the number of years since 1985 . Population \(W\) is modelled by the equation \(y=A \times(1.055)^x\). Population \(K\) is modelled by the equation \(y=B \times(0.97)^x\). Complete the table using the information provided. (3 marks) \begin{array}{|l|c|c|} --- 5 WORK AREA LINES (style=lined) ---
\hline
\rule{0pt}{2.5ex}\rule[-1ex]{0pt}{0pt}&\text {Population } W & \text {Population } K \\
\hline
\rule{0pt}{2.5ex}\text { Population in 1985 }\rule[-1ex]{0pt}{0pt} & A=34 & B=\ \ \ \ \ \\
\hline
\rule{0pt}{2.5ex}\text { Percentage yearly change in the population }\rule[-1ex]{0pt}{0pt} & & \\
\hline
\rule{0pt}{2.5ex}\text { Predicted population when } x=50 \rule[-1ex]{0pt}{0pt}& & 61 \\
\hline
\end{array}
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)
L&E, 2ADV E1 SM-Bank 2
The population of Indian Myna birds in a suburb can be described by the exponential function
`N = 35e^(0.07t)`
where `t` is the time in months.
- What will be the population after 2 years? (1 mark)
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- Draw a graph of the population. (2 marks)
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Show Answers Only
- `188\ text(birds)`
-
Show Worked Solution
i. `N = 35e^(0.07t)`
`text(Find)\ N\ text(when)\ \ t = 24:`
| `N` | `= 35e^(0.07 xx 24)` |
| `= 35e^(1.68)` | |
| `= 187.79…` | |
| `= 188\ text(birds)` |
ii.
Algebra, STD2 A4 2012 HSC 30c
In 2010, the city of Thagoras modelled the predicted population of the city using the equation
`P = A(1.04)^n`.
That year, the city introduced a policy to slow its population growth. The new predicted population was modelled using the equation
`P = A(b)^n`.
In both equations, `P` is the predicted population and `n` is the number of years after 2010.
The graph shows the two predicted populations.
- Use the graph to find the predicted population of Thagoras in 2030 if the population policy had NOT been introduced. (1 mark)
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- In each of the two equations given, the value of `A` is 3 000 000.
What does `A` represent? (1 mark)
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- The guess-and-check method is to be used to find the value of `b`, in `P = A(b)^n`.
(1) Explain, with or without calculations, why 1.05 is not a suitable first estimate for `b`. (1 mark)
(2) With `n = 20` and `P = 4\ 460\ 000`, use the guess-and-check method and the equation `P = A(b)^n` to estimate the value of `b` to two decimal places. Show at least TWO estimate values for `b`, including calculations and conclusions. (2 marks)
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- The city of Thagoras was aiming to have a population under 7 000 000 in 2050. Does the model indicate that the city will achieve this aim?
Justify your answer with suitable calculations. (2 marks)
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