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Vectors and matrices — Question 14

QCAA 2021, Paper 2 · 5 marks

Q14 · 2021 · Technology-activeSimple familiar5 marks

QUESTION 14 (5 marks)

The Tasmanian thornbill is a species of bird that has an average life span of three years. Female thornbills do not reproduce in their first year, but produce an average of four female offspring in each of their second and third years. The survival rate of each age group is estimated as 25% in their first year and 30% in their second year.
A Leslie matrix, L\mathbf L, modelling the population distribution of the Tasmanian thornbill, has been partially completed: L=[044x000y0].\mathbf L=\begin{bmatrix}0&4&4\\x&0&0\\0&y&0\end{bmatrix}.
a)
State the values of xx and yy.
[1 mark]
b)
At the start of 2021, a study began into the population of Tasmanian thornbills. The study estimated that the initial female population was 510 in their first year, 480 in their second year and 420 in their third year, and found that the ratio of male to female was approximately 1:21:2. Estimate the total population of Tasmanian thornbills at the start of 2025.
[4 marks]
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