QCE Mathematical Methods: Exam practice Q16 (2025, Paper 1)

Mathematical Methods — Question 16

QCAA 2025, Paper 1 · 5 marks

Q16 · 2025 · Technology-freeComplex familiar5 marks

QUESTION 16 (5 marks)

The energy produced by solar panels on a house is stored in a battery. Energy consumed in the house comes out of the battery.

The rate of energy produced by the solar panels during daylight hours on a particular day is approximately modelled by

R(t)=2.5\sin\left(\frac{\pi}{12}(t-6)\right),\qquad 6\le t\le18,

where

t

is the number of hours after midnight.

R(t)=0

for any hours after sunset and before sunrise.

The rate of energy consumed in the house is

\frac4\pi

units per hour. Determine the total energy change in the battery during daylight hours, assuming that sunrise is at 6:00 am and sunset is at 6:00 pm.

WORKED SOLUTION

5 marks

ANSWER

\frac{12}{\pi}

units.

Worked solution

Total energy produced:

\int_6^{18}2.5\sin\left(\frac\pi{12}(t-6)\right)\,dt =\left[-\frac{30}{\pi}\cos\left(\frac\pi{12}(t-6)\right)\right]_6^{18} =\frac{60}{\pi}.

Total energy consumed during the 12 daylight hours:

12\times\frac4\pi=\frac{48}{\pi}.

Therefore the total energy change is

\frac{60}{\pi}-\frac{48}{\pi}=\frac{12}{\pi}.

correctly integrates the energy produced term

[1 mark]

determines the energy produced between sunrise and sunset

[1 mark]

correctly determines a method to find the energy consumed between sunrise and sunset

[1 mark]

determines the energy consumed during daylight hours

[1 mark]

determines the total energy change between sunrise and sunset

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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