Mathematical Methods — Question 17

QCAA 2023, Paper 1 · 6 marks

Q17 · 2023 · Technology-freeComplex familiar6 marks

QUESTION 17 (6 marks)

A chemical is added to the water in a swimming pool at 10:00 am to prevent algae. The amount of chemical absorbed into the water over time

t

(hours) is represented by

A=10t^2-4t^3,\qquad0\le t\le1\frac23.

Determine the time of day when the rate of absorption of the chemical is at its maximum. Use calculus techniques to verify that your time corresponds to a maximum rate.

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WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

10:50 am.

Worked solution

The rate of absorption is

\frac{dA}{dt}=20t-12t^2.

The second derivative is

\frac{d^2A}{dt^2}=20-24t.

Setting this equal to zero gives

20-24t=0\Rightarrow t=\frac{20}{24}=\frac56\text{ hours}.

To verify this corresponds to a maximum rate,

\frac{d^3A}{dt^3}=-24,

which is negative, so the rate of absorption is a maximum. The time is

\frac56\times60=50

minutes after 10:00 am. The required time is **10:50 am**.

correctly determines the first derivative

[1 mark]

determines the second derivative

[1 mark]

equates the second derivative to zero

[1 mark]

determines time when second derivative is zero

[1 mark]

performs a calculus test to confirm the time corresponds to a maximum for dA/dt

[1 mark]

determines the time for maximum rate of absorption in minutes

[1 mark]

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Mathematical Methods — Question 17

QUESTION 17 (6 marks)

QCAA guide · typeset solution

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