Date | May 2019 | Marks available | 2 | Reference code | 19M.1.SL.TZ2.T_13 |
Level | Standard Level | Paper | Paper 1 (with calculator from previous syllabus) | Time zone | Time zone 2 |
Command term | Find and Justify | Question number | T_13 | Adapted from | N/A |
Question
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.
Find the value of a.
Find the time, in years, for the population to decrease to 20 turtles.
There is a number m beyond which the turtle population will not decrease.
Find the value of m. Justify your answer.
Markscheme
* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.
55=a×20+10 (M1)
Note: Award (M1) for correct substitution of zero and 55 into the function.
45 (A1) (C2)
[2 marks]
45×2−t+10⩽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]
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]