In a forest, the population of a species of mouse is falling by 2.7% each year. To increase the population scientists plan to release 30 mice into the forest at the end of March each year. a) $u_n$ is the estimated population of mice at the start of April, $n$ years after the population was first estimated. It is known that $u_n$ and $u_{n+1}$ satisfy the recurrence relation $u_{n+1}=au_n+b$. State the values of $a$ and $b$. The scientists continue to release this species of mouse each year. b) (i) Explain why the estimated population of mice will stabilise in the long term. (ii) Calculate the long term population to the nearest hundred.

Scottish Highers Maths Exponential and Logs: In a forest, the population of a

In a forest, the population of a species of mouse is falling by 2.7% each year - Scottish Highers Maths - Question 4 - 2019

Question 4

In a forest, the population of a species of mouse is falling by 2.7% each year. To increase the population scientists plan to release 30 mice into the forest at the ... show full transcript

Worked Solution & Example Answer:In a forest, the population of a species of mouse is falling by 2.7% each year - Scottish Highers Maths - Question 4 - 2019

Step 1

State the values of a and b

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Answer

From the problem, we know that the mouse population decreases by 2.7% each year. Therefore, the factor by which the population decreases can be represented as:

$a = 1 - 0.027 = 0.973$

Additionally, 30 mice are released each year, which means:

$b = 30$

So, the values are:

$a = 0.973$
$b = 30$

Step 2

Explain why the estimated population of mice will stabilise in the long term

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Answer

The estimated population of mice will stabilise in the long term because the recurrence relation is linear and involves a constant addition of 30 mice to the decreasing population. As the population falls by a fixed percentage (2.7%), the balance between losses due to natural decline and gains from the release of new mice (30 each year) will ultimately result in a stable long-term population.

Step 3

Calculate the long term population to the nearest hundred

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Answer

To find the long-term population, we need to determine the limit of the recurrence relation as $n$ approaches infinity. We can denote the long-term population as:

$L = 0.973L + 30$

Rearranging gives:

$L - 0.973L = 30$ $0.027L = 30$ $L = \frac{30}{0.027} \approx 1111.11$

Rounding to the nearest hundred, the long-term population stabilises at approximately 1100.

In a forest, the population of a species of mouse is falling by 2.7% each year - Scottish Highers Maths - Question 4 - 2019

Worked Solution & Example Answer:In a forest, the population of a species of mouse is falling by 2.7% each year - Scottish Highers Maths - Question 4 - 2019

State the values of a and b

Explain why the estimated population of mice will stabilise in the long term

Calculate the long term population to the nearest hundred

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