7A closed cylinder has radius $2\text{ cm}$ and height $7\text{ cm}$. What is its total surface area, to 2 decimal places?L7
A $56.55\text{ cm}^2$
B $87.96\text{ cm}^2$
C $113.10\text{ cm}^2$
D $226.19\text{ cm}^2$
C — $113.10\text{ cm}^2$. $SA = 2\pi r^2 + 2\pi r h = 2\pi(4) + 2\pi(2)(7) = 8\pi + 28\pi = 36\pi \approx 113.10\text{ cm}^2$.
8A square pyramid has a base of $6\text{ cm} \times 6\text{ cm}$ and slant height $5\text{ cm}$. What is its total surface area?L8
A $60\text{ cm}^2$
B $96\text{ cm}^2$
C $156\text{ cm}^2$
D $120\text{ cm}^2$
B — $96\text{ cm}^2$. Base $= 36\text{ cm}^2$. Four triangular faces: $4 \times \tfrac{1}{2} \times 6 \times 5 = 60\text{ cm}^2$. Total $= 36 + 60 = 96\text{ cm}^2$.
9A cylindrical tank has radius $3\text{ m}$ and height $5\text{ m}$. How many litres of water can it hold? (Recall: $1\text{ m}^3 = 1000\text{ L}$, answer to nearest whole litre.)L9
A $45\,000\text{ L}$
B $141\,372\text{ L}$
C $471\,239\text{ L}$
D $282\,743\text{ L}$
B — $141\,372\text{ L}$. $V = \pi r^2 h = \pi(9)(5) = 45\pi \approx 141.372\text{ m}^3 = 141\,372\text{ L}$.
10A hemisphere has radius $6\text{ cm}$. What is its volume, to 2 decimal places?L10
15In a right-angled triangle, the side opposite $\theta$ is $11\text{ cm}$ and the hypotenuse is $14\text{ cm}$. Find $\theta$ to the nearest minute.L15
17From the top of a $45\text{ m}$ lighthouse, the angle of depression to a boat is $18°$. How far is the boat from the base, to 1 decimal place?L16
A $14.6\text{ m}$
B $47.3\text{ m}$
C $138.5\text{ m}$
D $146.4\text{ m}$
C — $138.5\text{ m}$. The angle of depression equals the angle of elevation from the boat. $\tan 18° = \dfrac{45}{d} \Rightarrow d = \dfrac{45}{\tan 18°} \approx \dfrac{45}{0.3249} \approx 138.5\text{ m}$.
18A ship sails on a bearing of $070°$. What is the back bearing (the bearing back to the starting point)?L17
A $070°$
B $110°$
C $250°$
D $290°$
C — $250°$. Since the bearing is less than $180°$, back bearing $= 070° + 180° = 250°$.
19A bushwalker travels N$35°$E for $8\text{ km}$. How far north has she travelled, to 2 decimal places?L17
20Point C is on a bearing of $040°$ from point A. What is the bearing from C back to A?L17
A $040°$
B $140°$
C $220°$
D $310°$
C — $220°$. $040° < 180°$, so back bearing $= 040° + 180° = 220°$.
21A 2.5 kW air conditioner runs for 4 hours per day. How much energy does it use in one day?L18
A $2.5\text{ kWh}$
B $6.5\text{ kWh}$
C $10\text{ kWh}$
D $0.625\text{ kWh}$
C — $10\text{ kWh}$. $E = P \times t = 2.5\text{ kW} \times 4\text{ h} = 10\text{ kWh}$.
22The trapezoidal rule is applied to a shape with $h = 3\text{ m}$, first width $d_f = 4\text{ m}$, middle width $d_m = 7\text{ m}$, and last width $d_l = 4\text{ m}$. What is the estimated area?L19
23A train departs at $11{:}48\text{ pm}$ and arrives at $2{:}15\text{ am}$ the following morning. What is the journey time?L20
A $2\text{ h } 15\text{ min}$
B $2\text{ h } 27\text{ min}$
C $3\text{ h } 33\text{ min}$
D $14\text{ h } 27\text{ min}$
B — $2\text{ h } 27\text{ min}$. Count up: 11:48 pm → midnight = 12 min; midnight → 2:15 am = 2 h 15 min. Total $= 12 + 135 = 147\text{ min} = 2\text{ h } 27\text{ min}$.
24When it is $8{:}00\text{ pm}$ in Sydney (AEST, UTC$+10$), what time is it in Dubai (UTC$+4$)?L21
A $2{:}00\text{ am}$
B $10{:}00\text{ pm}$
C $2{:}00\text{ pm}$
D $6{:}00\text{ pm}$
C — $2{:}00\text{ pm}$. Sydney is 6 hours ahead of Dubai (UTC$+10$ vs UTC$+4$). So Dubai time $= 8{:}00\text{ pm} - 6\text{ h} = 2{:}00\text{ pm}$.
25City A is at longitude $150°\text{E}$ and City B is at longitude $30°\text{E}$. Using the rule that Earth rotates $15°$ per hour, what is the time difference between the two cities?L22
A $6\text{ hours}$
B $4\text{ hours}$
C $8\text{ hours}$
D $12\text{ hours}$
C — $8\text{ hours}$. $\Delta\lambda = 150° - 30° = 120°$. Time difference $= 120° \div 15°/\text{h} = 8\text{ hours}$. City A is 8 hours ahead of City B.
Part B — Short Answer (show all working)
1L5 & L6
A circular swimming pool has diameter $8\text{ m}$.
(a) Find the circumference of the pool, to 2 decimal places.
(b) Find the area of the pool, to 2 decimal places.
(c) A circular fence is built $1\text{ m}$ outside the pool. Find the area of the annular gap between pool edge and fence, to 2 decimal places.
(c) $D = S \times T = 65 \times 2.5 = 162.5\text{ km}$
4L14 & L16
From point A on level ground, the angle of elevation to the top T of a building is $42°$. From point B, which is $20\text{ m}$ closer to the building than A, the angle of elevation to T is $58°$. Find the height of the building, to 1 decimal place.
(a) Let $d$ = horizontal distance from B to the building. Write two equations for $h$ (the height) in terms of $d$.
(b) Solve for $d$, then find $h$.
(a) From A (distance $d + 20$ from building): $h = (d+20)\tan 42°$
From B (distance $d$ from building): $h = d\tan 58°$
A farmer uses a 3.5 kW water pump to irrigate a paddock. The pump runs for 4 hours each day for 7 days.
(a) Calculate the total energy used by the pump over the 7 days, in kWh.
(b) Electricity costs $0.25 per kWh. How much does the irrigation cost for the week?
(c) The paddock has an irregular shape. Five cross-sectional widths are measured 8 m apart: 6 m, 10 m, 14 m, 12 m, 8 m. Use the trapezoidal rule to estimate the paddock's area.
(a) Total time $= 4 \times 7 = 28\text{ h}$. $E = P \times t = 3.5 \times 28 = 98\text{ kWh}$
City A is at longitude $120°\text{E}$ (UTC$+8$) and City B is at longitude $165°\text{E}$ (UTC$+11$). A train timetable for City B shows: depart 07:42, arrive Central 10:18, arrive Eastport 12:05.
(a) How long is the train journey from the departure station to Eastport?
(b) A video call is scheduled for 09:00 in City A (UTC$+8$). What time is it in City B (UTC$+11$) at this moment?
(c) Using the longitude rule ($15°$ per hour), calculate the theoretical time difference between City A and City B. Does it agree with the UTC offset difference?
(a) Depart 07:42, arrive Eastport 12:05.
$12{:}05 - 07{:}42 = 4\text{ h } 23\text{ min}$
(b) City B is UTC$+11$, City A is UTC$+8$ — City B is 3 hours ahead.