Differentiation of exponential and logarithmic functions — Question 105

Original QCE Vault practice · 5 marks

Q105 · Practice questionTechnology-activeComplex unfamiliar5 marks

QUESTION 105 (5 marks)

The concentration of a substance is modelled by

C(t)=t^2e^{-t/3}

for

t\ge0

. Determine the time at which

C

is maximised. Then determine

C(3)

as a percentage of this maximum value.

WORKED SOLUTION

5 marks

ANSWER

The maximum occurs at

t=6

. Also,

C(3)/C(6)=e/4

, so

C(3)

is approximately

68.0\%

of the maximum.

Worked solution

C'(t)=e^{-t/3}(2t-t^2/3)=te^{-t/3}(2-t/3)

. The positive interior stationary point is

t=6

, where the derivative changes from positive to negative.

C(3)=9e^{-1}

and

C(6)=36e^{-2}

, giving the ratio

e/4\approx0.6796

Differentiates

C(t)

correctly.

[1 mark]

Finds the interior stationary point

t=6

[1 mark]

Justifies that

t=6

gives a maximum.

[1 mark]

Forms the ratio

C(3)/C(6)

[1 mark]

Obtains approximately

68.0\%

[1 mark]

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

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