Solved: Little Green island originally had no turtles. After 55 turtles were introduced... [IB Mathematics Applications & Interpretation (Math AI)]

Little Green island originally had no turtles. After 55 turtles were introduced to the island, their population is modelled by N ( t ) = a × 2 − t + 10 , t ⩾ 0 , where a is a constant and t is the time in years since the turtles were introduced.

Question

SLPaper 1

Little Green island originally had no turtles. After 55 turtles were introduced to the island, their population is modelled by

$N\left( t \right) = a \times {2^{ - t}} + 10{\text{,}}\,\,\,t \geqslant 0{\text{,}}$

where $a$ is a constant and $t$ is the time in years since the turtles were introduced.

Find the value of $a$ .

[2]

Verified

Solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$55 = a \times {2^0} + 10$ (M1)

Note: Award (M1) for correct substitution of zero and 55 into the function.

45 (A1) (C2)

[2 marks]

Find the time, in years, for the population to decrease to 20 turtles.

[2]

Verified

Solution

$45 \times {2^{ - t}} + 10 \leqslant 20$ (M1)

Note: Award (M1) for comparing correct expression involving 20 and their 45. Accept an equation.

$t = 2.17$ (2.16992…) (A1)(ft) (C2)

Note: Follow through from their part (a), but only if positive.
Answer must be in years; do not accept months for the final (A1).

[2 marks]

There is a number $m$ beyond which the turtle population will not decrease.

Find the value of $m$ . Justify your answer.

[2]

Verified

Solution

$m =$ 10 (A1)

because as the number of years increases the number of turtles approaches 10 (R1) (C2)

Note:Award (R1) for a sketch with an asymptote at approximately $y = 10$ ,
OR for table with values such as 10.003 and 10.001 for $t = 14$ and $t = 15$ , for example,
OR when $t$ approaches large numbers $y$ approaches 10. Do not award (A1)(R0).

[2 marks]

Little Green island originally had no turtles. After 55 turtles were introduced to the island, their population is modelled by N ( t ) = a × 2 − t + 10 , t ⩾ 0 , where a is a constant and t is the time in years since the turtles were introduced.

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