Population Growth Model

00:00 6 marks

The population, \(P\), of a town \(t\) years after the beginning of 2010 is modelled by the function \(P(t) = 15000 \times (1.025)^t\).

(a) Write down the population of the town at the beginning of 2010. [1]
(b) Write down the annual percentage growth rate of the population. [1]
(c) Calculate the population of the town at the beginning of 2025. [2]
(d) Find the year in which the population will first exceed 25000. [2]

Part a

2 minutes 1 marks

Question

Write down the population of the town at the beginning of 2010. [1]

Hint Hint

Hint

Substitute \(t=0\) into \(P(t)=15000 \times (1.025)^t\).

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15000

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Part b

1 minutes 1 marks

Question

Write down the annual percentage growth rate of the population. [1]

Hint Hint

Hint

The growth factor \(1.025\) means the growth rate is \(1.025 – 1\) expressed as a percentage.

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2.5%

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Part c

3 minutes 2 marks

Question

Calculate the population of the town at the beginning of 2025. [2]

Hint Hint

Hint

For 2025, we have \(t = 2025 – 2010 = 15\). Substitute this into the function \(P(t)=15000 \times (1.025)^t\).

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Approximately 21719

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Part d

4 minutes 2 marks

Question

Find the year in which the population will first exceed 25000. [2]

Hint Hint

Hint

Set up the inequality \(15000 \times (1.025)^t > 25000\), divide by 15000, and solve using logarithms.

Answer Answer
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2031

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