19M.2.SL.TZ2.S_5

pestleMathematicsAASLPaper 219M· sl-5-6-differentiating-polynomials-n-e-q-chain-product-and-quotient-rulessource ↗

The population of fish in a lake is modelled by the function

$f\left( t \right) = \frac{{1000}}{{1 + 24{{\text{e}}^{ - 0.2t}}}}$ , 0 ≤ $t$ ≤ 30 , where $t$ is measured in months.

Find the population of fish at $t$ = 10.

[2]

Find the rate at which the population of fish is increasing at $t$ = 10.

[2]

Find the value of $t$ for which the population of fish is increasing most rapidly.

[2]

Markscheme / solution

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

valid approach (M1)

eg $f$ (10)

235.402

235 (fish) (must be an integer) A1 N2

[2 marks]

recognizing rate of change is derivative (M1)

eg rate = ${f'}$ , ${f'}$ (10) , sketch of ${f'}$ , 35 (fish per month)

35.9976

36.0 (fish per month) A1 N2

[2 marks]

valid approach (M1)

eg maximum of ${f'}$ , ${f''}$ = 0

15.890

15.9 (months) A1 N2

[2 marks]

Examiners’ report

[N/A]