2024 VCE Maths Methods Mini Test 4

Question 1 [2024 Exam 2 Section B Q2]

A model for the temperature in a room, in degrees Celsius, is given by

a. Express the derivative \( f'(t) \) as a hybrid function. 2 marks

b. Find the average rate of change in temperature predicted by the model between \( t = 0 \) and \( t = \frac{1}{2} \). Give your answer in degrees Celsius per hour. 1 mark

c. Another model for the temperature in the room is given by \( g(t) = 22 - 10e^{-6t},\ t \geq 0 \).

i. Find the derivative \( g'(t) \). 1 mark

ii. Find the value of \( t \) for which \( g'(t) = 10 \). Give your answer correct to three decimal places. 1 mark

d. Find the time \( t \in (0, 1) \) when the temperatures predicted by the models \( f \) and \( g \) are equal. Give your answer correct to two decimal places. 1 mark

e. Find the time \( t \in (0, 1) \) when the difference between the temperatures predicted by the two models is the greatest. Give your answer correct to two decimal places. 1 mark

f. The amount of power, in kilowatts, used by the heater \( t \) hours after it is switched on, can be modelled by the continuous function \( p \), whose graph is shown below.

\\[ p(t) = \begin{cases} 1.5 & 0 \leq t \leq 0.4 \\ 0.3 + Ae^{-10t} & t > 0.4 \end{cases} \]

The amount of energy used by the heater, in kilowatt hours, can be estimated by evaluating the area between the graph of \( y = p(t) \) and the \( t \)-axis.

i. Given that \( p(t) \) is continuous for \( t \geq 0 \), show that \( A = 1.2e^4 \). 1 mark

ii. Find how long it takes, after the heater is switched on, until the heater has used 0.5 kilowatt hours of energy. Give your answer in hours. 1 mark

iii. Find how long it takes, after the heater is switched on, until the heater has used 1 kilowatt hour of energy. Give your answer in hours, correct to two decimal places. 2 marks