Further integration — Question 130

Original QCE Vault practice · 6 marks

Q130 · Practice questionTechnology-activeComplex unfamiliar6 marks

QUESTION 130 (6 marks)

A tank initially contains

V_0

litres of water. For

0\le t\le10

minutes, water enters at a rate

4e^{-0.2t}

L/min while a pump removes water at a constant rate of

2

L/min. Determine the minimum value of

V_0

that guarantees the tank does not become empty during the 10-minute interval.

WORKED SOLUTION

6 marks

ANSWER

V_0=20e^{-2}\approx2.71

Worked solution

The net rate is

R(t)=4e^{-0.2t}-2

. It is initially positive and becomes negative after

t=5\ln2

. The cumulative change by time

t

20(1-e^{-0.2t})-2t

. Its smallest value on

[0,10]

is at

t=10

, where it equals

-20e^{-2}

. Therefore the minimum initial volume is

20e^{-2}\approx2.71

Forms the net rate of change.

[1 mark]

Integrates to obtain cumulative change.

[1 mark]

Analyses where the cumulative volume can be minimal.

[1 mark]

Evaluates the cumulative change at the relevant endpoint.

[1 mark]

Relates the deficit to the required initial volume.

[1 mark]

States the minimum volume with units.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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