Mathematical Methods — Question 17

QCAA 2022, Paper 2 · 4 marks

Q17 · 2022 · Technology-activeComplex unfamiliar4 marks

QUESTION 17 (4 marks)

A snail is travelling along a straight path from point

A

. The snail’s velocity (cm min

^{-1}

) is modelled by

v(t)=1.4\ln(1+t^2)

, where

t

is time (in minutes) for

0\le t\le15

. An ant passes point

A

12 minutes after the snail and follows the snail’s path. The ant moves with a constant acceleration of 2 cm min

^{-2}

and passes the snail at

t=15

minutes. Determine the ant’s velocity at point

A

WORKED SOLUTION

4 marks

ANSWER

22.3477

cm min

^{-1}

along the ant’s path.

Worked solution

The snail’s total displacement is

\int_0^{15}1.4\ln(1+t^2)\,dt=76.0431\text{ cm}.

For the ant,

v(t)=2t+c

. Because the ant travels the same displacement from

t=12

t=15

\int_{12}^{15}(2t+c)\,dt=76.0431.

Solving gives

c=-1.6523

. Therefore at point

A

(

t=12

v(12)=24-1.6523=22.3477\text{ cm min}^{-1}.

correctly determines the total displacement of the snail

[1 mark]

establishes an equation linking the ant and the snail

[1 mark]

determines constant

[1 mark]

determines velocity of ant

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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