Differentiation of exponential and logarithmic functions — Question 154

Original QCE Vault practice · 4 marks

Q154 · Practice questionTechnology-activeComplex familiar4 marks

QUESTION 154 (4 marks)

The temperature of a drink is modelled by

T(t)=18+72e^{-0.2t}

degrees Celsius, where

t

is the number of minutes after the drink is poured.

(a) Determine the instantaneous rate of change of the temperature at

t=4

(b) Determine when the temperature first reaches

30^\circ\mathrm C

WORKED SOLUTION

4 marks

ANSWER

(a)

T'(4)\approx -6.47\ ^\circ\mathrm C/\mathrm{min}

; (b)

t=5\ln 6\approx 8.96

min.

Worked solution

Differentiate to obtain

T'(t)=-14.4e^{-0.2t}

. Substitution gives the cooling rate at

t=4

. For

T=30

, solve

12=72e^{-0.2t}

, so

e^{-0.2t}=1/6

and

t=5\ln6

Differentiates the exponential model correctly.

[1 mark]

Evaluates the rate at

t=4

[1 mark]

Forms the exponential equation for

T=30

[1 mark]

Solves for the time.

[1 mark]

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

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