A population, P, is to be modelled using the function $P = 2000(1.2)^t$, where $t$ is the time in years. (a) What is the initial population? (b) Find the population after 5 years. (c) On the axes below, draw the graph of the population against time, showing the points at $t = 0$ and at $t = 5$.

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A population, P, is to be modelled using the function $P = 2000(1.2)^t$, where $t$ is the time in years - HSC - SSCE Mathematics Standard - Question 24 - 2021 - Paper 1

Question 24

A population, P, is to be modelled using the function $P = 2000(1.2)^t$, where $t$ is the time in years. (a) What is the initial population? (b) Find the populatio... show full transcript

Worked Solution & Example Answer:A population, P, is to be modelled using the function $P = 2000(1.2)^t$, where $t$ is the time in years - HSC - SSCE Mathematics Standard - Question 24 - 2021 - Paper 1

Step 1

What is the initial population?

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Answer

To find the initial population, we evaluate the function at $t = 0$ :

P = 2000(1.2)^0 \\ = 2000(1) \\ = 2000

Thus, the initial population is 2000.

Step 2

Find the population after 5 years.

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Answer

To determine the population after 5 years, we substitute $t = 5$ into the function:

P = 2000(1.2)^5 \\ = 2000(2.48832) \\ ext{(calculating results in } 4976.64)

Rounding to the nearest integer, the population after 5 years is 4977.

Step 3

On the axes below, draw the graph of the population against time, showing the points at $t = 0$ and at $t = 5$.

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Answer

To graph the population against time:

Plot the point at $t = 0$ , $P = 2000$ .
Plot the point at $t = 5$ , $P = 4977$ .
Connect these points with a smooth curve that represents the exponential growth, ensuring the curve has a positive slope.

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