Year 12 · Measurement
Quick tips — memory joggers
Rates
Consumption & energy rates
Ratios
Scale, area & volume
Level 1 · Fluency
A tap fills a bucket at 4 litres per minute. How much water flows in 7 minutes?
28 litres
\(4 \times 7 =\) 28 litres.
A car travels at a steady 90 km/h. How far does it travel in 3 hours?
270 km
Distance \(=\) speed \(\times\) time \(= 90 \times 3 =\) 270 km.
A garden hose delivers 12 litres per minute. How much water is used in 9 minutes?
108 litres
\(12 \times 9 =\) 108 litres.
Level 2 · Application
A cyclist rides 12 km at 24 km/h, then 9 km at 18 km/h. Calculate the total time taken.
1 hour
Time \(= \dfrac{12}{24} + \dfrac{9}{18} = 0.5 + 0.5 =\) 1 hour.
A train covers 210 km in 2.5 hours. Travelling at the same average speed, how far will it go in 4 hours?
336 km
Speed \(= \dfrac{210}{2.5} = 84\) km/h; distance \(= 84 \times 4 =\) 336 km.
A dripping tap loses 3 mL every 5 seconds. How many litres are wasted in 24 hours? (1 L \(=\) 1000 mL.)
51.84 litres
Rate \(= \dfrac{3}{5} = 0.6\) mL/s; in 24 h there are \(24 \times 3600 = 86\,400\) s; water \(= 0.6 \times 86\,400 = 51\,840\) mL \(=\) 51.84 litres.
Level 3 · Further Application
A printer prints 18 pages per minute. A job has 450 pages.
9.375 min \(\approx\) 9 min 23 s
(a) \(450 \div 18 = 25\) minutes.
(b) Combined rate = \(18 + 30 = 48\) pages/min; time = \(450 \div 48 =\) 9.375 min \(\approx\) 9 min 23 s.
A pump fills a pool at 25 litres per minute.
150 min (2.5 h)
(a) \(9000 \div 25 = 360\) minutes (6 hours).
(b) Combined rate \(= 25 + 35 = 60\) L/min; time \(= 9000 \div 60 =\) 150 min (2.5 h).
A worker is paid $27.50 per hour.
$330.00
(a) \(27.50 \times 7.5 =\) $206.25.
(b) Overtime rate \(= 27.50 \times 1.5 = 41.25\); overtime pay \(= 41.25 \times 3 = 123.75\); total \(= 206.25 + 123.75 =\) $330.00.
Level 1 · Fluency
Which is better value: 2 kg for $6 or 5 kg for $12.50?
5 kg
$6/2 = $3.00/kg vs $12.50/5 = $2.50/kg. The 5 kg pack is cheaper per kg.
Who runs faster: runner A who covers 400 m in 80 s, or runner B who covers 500 m in 90 s?
Runner B
A: \(\dfrac{400}{80} = 5\) m/s; B: \(\dfrac{500}{90} \approx 5.56\) m/s. Runner B is faster.
Which is better value: 300 g for $4.50 or 500 g for $7.00?
500 g pack
$4.50/300 g \(=\) $1.50/100 g vs $7.00/500 g \(=\) $1.40/100 g. The 500 g pack is cheaper per 100 g.
Level 2 · Application
A person’s target heart-rate zone is 120–150 beats/min. They count 38 beats in 15 seconds. Determine whether they are in the zone.
slightly above
Rate = \(38 \times \dfrac{60}{15} = 152\) beats/min. This is just above 150, so they are slightly above the target zone.
Car A travels 480 km on 40 L of fuel; Car B travels 620 km on 50 L. Which car is more fuel-efficient (more km per litre)?
Car B (12.4 km/L)
A: \(\dfrac{480}{40} = 12\) km/L; B: \(\dfrac{620}{50} = 12.4\) km/L. Car B (12.4 km/L) is more efficient.
A resting patient takes 27 breaths in 90 seconds. A normal resting rate is 12–20 breaths/min. Determine whether the patient is within the normal range.
within normal range
Rate \(= 27 \times \dfrac{60}{90} = 18\) breaths/min, which lies in 12–20, so the patient is within the normal range.
Level 3 · Further Application
Two brands of juice are offered.
| Brand | Size | Price |
|---|---|---|
| A | 1.5 L | $4.20 |
| B | 2 L | $5.40 |
(a) A: \(\dfrac{4.20}{1.5} =\) $2.80/L. B: \(\dfrac{5.40}{2} =\) $2.70/L.
(b) Brand B is better value (lower cost per litre).
Coffee is sold in three sizes.
| Size | Price |
|---|---|
| 250 g | $7.50 |
| 500 g | $13.00 |
| 1 kg | $24.00 |
1 kg pack ($2.40/100 g)
(a) 250 g: \(\dfrac{7.50}{2.5} =\) $3.00/100 g; 500 g: \(\dfrac{13.00}{5} =\) $2.60/100 g; 1 kg: \(\dfrac{24.00}{10} =\) $2.40/100 g.
(b) The 1 kg pack ($2.40/100 g) is best value.
Worker A packs 45 boxes in 30 minutes; worker B packs 70 boxes in 50 minutes.
Worker A (90 boxes/h)
(a) A: \(45 \times \dfrac{60}{30} = 90\) boxes/h; B: \(70 \times \dfrac{60}{50} = 84\) boxes/h.
(b) Worker A (90 boxes/h) is faster.
Level 1 · Fluency
A 60 W globe uses how many joules of energy each second?
60 joules per second (1 W = 1 J/s).
A 25 W device uses how many joules of energy in one second?
25 joules (since 1 W = 1 J/s).
How many joules of energy does a 100 W bulb use in 10 seconds?
1000 joules
100 W = 100 J/s; energy \(= 100 \times 10 =\) 1000 joules.
Level 2 · Application
A 1500 W heater runs for 2 hours. Calculate the energy used in kilowatt-hours (kWh).
3 kWh
1500 W = 1.5 kW; energy = \(1.5 \times 2 =\) 3 kWh.
A 2400 W kettle runs for 15 minutes. Calculate the energy used in kilowatt-hours (kWh).
0.6 kWh
2400 W = 2.4 kW; 15 min = 0.25 h; energy \(= 2.4 \times 0.25 =\) 0.6 kWh.
An 800 W microwave runs for 90 seconds. Calculate the energy used in joules.
72 000 joules
800 W = 800 J/s; energy \(= 800 \times 90 =\) 72 000 joules (72 kJ).
Level 3 · Further Application
A 2000 W air-conditioner runs for 5 hours a day.
$84
(a) 2 kW \(\times\) 5 h = 10 kWh/day.
(b) \(10 \times 30 = 300\) kWh; cost = \(300 \times 0.28 =\) $84.
A 3500 W pool pump runs for 6 hours a day.
$604.80
(a) 3.5 kW \(\times\) 6 h = 21 kWh/day.
(b) \(21 \times 90 = 1890\) kWh; cost = \(1890 \times 0.32 =\) $604.80.
A heater is rated at 2000 W.
(a) 120 000 J (b) 8 kWh
(a) 2000 W = 2000 J/s; \(2000 \times 60 =\) 120 000 J.
(b) 2000 W = 2 kW; \(2 \times 4 =\) 8 kWh.
Level 1 · Fluency
Appliance A uses 400 kWh/year and appliance B uses 550 kWh/year. Which is cheaper to run?
Appliance A (it uses less energy per year).
TV A uses 120 kWh/year and TV B uses 95 kWh/year. Which is cheaper to run?
TV B (it uses less energy per year).
An appliance uses 250 kWh in a year. At 30c/kWh, calculate the annual running cost.
$75/year
Cost \(= 250 \times 0.30 =\) $75/year.
Level 2 · Application
Fridge A uses 380 kWh/year; Fridge B uses 520 kWh/year. At 30c/kWh, calculate the annual running-cost difference.
$42/year
Difference \(= (520 - 380) \times 0.30 = 140 \times 0.30 =\) $42/year.
Dryer A uses 480 kWh/year; Dryer B uses 610 kWh/year. At 29c/kWh, calculate the annual running-cost difference.
$37.70/year
Difference \(= (610 - 480) \times 0.29 = 130 \times 0.29 =\) $37.70/year.
An old heater uses 1500 kWh/year; a new model uses 900 kWh/year. At 31c/kWh, calculate the annual saving from the new model.
$186/year
Saving \(= (1500 - 900) \times 0.31 = 600 \times 0.31 =\) $186/year.
Level 3 · Further Application
Fridge A costs $900 and uses 380 kWh/year; Fridge B costs $700 and uses 520 kWh/year. Electricity is 30c/kWh.
(a) A: \(900 + 5 \times 380 \times 0.30 = 900 + 570 =\) $1470. B: \(700 + 5 \times 520 \times 0.30 = 700 + 780 =\) $1480.
(b) Fridge A is cheaper overall by $10.
Washer A costs $650 and uses 300 kWh/year; Washer B costs $500 and uses 460 kWh/year. Electricity is 30c/kWh.
Washer A ($1190 vs $1328)
(a) A: \(650 + 6 \times 300 \times 0.30 = 650 + 540 =\) $1190; B: \(500 + 6 \times 460 \times 0.30 = 500 + 828 =\) $1328.
(b) Washer A is cheaper overall by $138.
Air-conditioner A costs $1200 and uses 700 kWh/year; unit B costs $900 and uses 1000 kWh/year. Electricity is 33c/kWh.
Unit A ($2124 vs $2220)
(a) A: \(1200 + 4 \times 700 \times 0.33 = 1200 + 924 =\) $2124; B: \(900 + 4 \times 1000 \times 0.33 = 900 + 1320 =\) $2220.
(b) Unit A is cheaper overall by $96.
Level 1 · Fluency
Name one household system to which energy-efficiency ratings commonly apply.
Any of: hot-water systems, heating/cooling, insulation, lighting, or appliances.
State one benefit to a homeowner of building a more energy-efficient home.
Any of: lower electricity/running costs, greater comfort, or higher resale value.
Name the NSW planning scheme that sets sustainability and energy-efficiency targets for new homes.
BASIX (the Building Sustainability Index).
Level 2 · Application
A more efficient hot-water system costs $800 more but saves 900 kWh/year. At 28c/kWh, calculate the annual saving.
$252/year
Annual saving = \(900 \times 0.28 =\) $252/year.
Adding ceiling insulation costs $1200 and cuts heating energy by 1500 kWh/year. At 30c/kWh, calculate the annual saving.
$450/year
Annual saving \(= 1500 \times 0.30 =\) $450/year.
A rooftop solar system reduces grid electricity use by 3200 kWh/year. At 29c/kWh, calculate the annual saving.
$928/year
Annual saving \(= 3200 \times 0.29 =\) $928/year.
Level 3 · Further Application
The efficient hot-water system above costs $800 more up-front and saves $252 per year.
(a) \(800 \div 252 \approx 3.2\) years.
(b) It reduces energy use and greenhouse-gas emissions (environmental benefit).
Ceiling insulation costs $1200 and saves $450 per year on heating.
(a) ≈ 2.7 years (b) $3300
(a) \(1200 \div 450 \approx\) 2.7 years.
(b) \(10 \times 450 - 1200 = 4500 - 1200 =\) $3300.
A rooftop solar system costs $6000 and saves $928 per year on electricity.
(a) \(6000 \div 928 \approx 6.5\) years.
(b) It reduces reliance on fossil-fuel electricity and lowers greenhouse-gas emissions.
Level 1 · Fluency
A car uses 8 L/100 km. How much fuel is needed for 250 km?
20 litres
\(\dfrac{8}{100} \times 250 =\) 20 litres.
A car uses 7 L/100 km. How much fuel is needed for 400 km?
28 litres
\(\dfrac{7}{100} \times 400 =\) 28 litres.
A van uses 11 L/100 km. How much fuel is needed for 150 km?
16.5 litres
\(\dfrac{11}{100} \times 150 =\) 16.5 litres.
Level 2 · Application
A car with fuel consumption 6.7 L/100 km travels 1560 km. At $1.45/L, calculate the total fuel cost.
$151.55
Fuel = \(\dfrac{6.7}{100} \times 1560 = 104.52\) L; cost = \(104.52 \times 1.45 =\) $151.55.
A car with fuel consumption 9.2 L/100 km travels 640 km. At $1.75/L, calculate the total fuel cost.
$103.04
Fuel \(= \dfrac{9.2}{100} \times 640 = 58.88\) L; cost \(= 58.88 \times 1.75 =\) $103.04.
A motorbike uses 4.5 L/100 km and has a 18 L tank. How far can it travel on a full tank?
400 km
Distance \(= 18 \div \dfrac{4.5}{100} = 18 \times \dfrac{100}{4.5} =\) 400 km.
Level 3 · Further Application
Car A uses 6.2 L/100 km and Car B uses 9.1 L/100 km. Both travel 14 000 km/year, with fuel at $1.90/L.
$771.40/year
(a) A: \(\dfrac{6.2}{100} \times 14\,000 \times 1.90 =\) $1649.20. B: \(\dfrac{9.1}{100} \times 14\,000 \times 1.90 =\) $2420.60.
(b) Saving = $771.40/year.
Car A uses 5.8 L/100 km and Car B uses 8.4 L/100 km. Both travel 18 000 km/year, with fuel at $1.85/L.
$865.80/year
(a) A: \(\dfrac{5.8}{100} \times 18\,000 \times 1.85 =\) $1931.40; B: \(\dfrac{8.4}{100} \times 18\,000 \times 1.85 =\) $2797.20.
(b) Saving = $865.80/year.
A 1800 km trip is planned. Car A uses 6.5 L/100 km and Car B uses 8 L/100 km, with fuel at $1.95/L.
$52.65
(a) A: \(\dfrac{6.5}{100} \times 1800 \times 1.95 =\) $228.15; B: \(\dfrac{8}{100} \times 1800 \times 1.95 =\) $280.80.
(b) Saving = $52.65.
Level 1 · Fluency
A recipe mixes flour and sugar in the ratio \(3:1\). For 3 cups of flour, how much sugar is needed?
1 cup of sugar (the ratio \(3:1\) means 1 part sugar for every 3 parts flour).
Mortar mixes cement and sand in the ratio \(1:4\). For 2 buckets of cement, how much sand is needed?
8 buckets of sand
The ratio \(1:4\) means 4 parts sand per 1 part cement, so \(2 \times 4 =\) 8 buckets of sand.
A cordial is mixed with water in the ratio water : syrup \(= 5:1\). For 250 mL of syrup, how much water is needed?
1250 mL
5 parts water per 1 part syrup, so \(5 \times 250 =\) 1250 mL.
Level 2 · Application
To estimate fish numbers, 45 fish are tagged and released. Later, a sample of 60 fish contains 9 tagged. Estimate the total fish population \(N\) using \(N \approx \dfrac{\text{tagged} \times \text{sample}}{\text{recaptured}}\).
300 fish
\(N \approx \dfrac{45 \times 60}{9} = \dfrac{2700}{9} =\) 300 fish.
To estimate a bird population, 80 birds are tagged. Later a sample of 120 birds contains 16 tagged. Estimate the population \(N\) using \(N \approx \dfrac{\text{tagged} \times \text{sample}}{\text{recaptured}}\).
600 birds
\(N \approx \dfrac{80 \times 120}{16} = \dfrac{9600}{16} =\) 600 birds.
A fertiliser mixes nitrogen : phosphorus : potassium in the ratio \(4:1:2\). A 21 kg bag is made. How much nitrogen does it contain?
12 kg
Total parts \(= 4 + 1 + 2 = 7\); one part \(= 21 \div 7 = 3\) kg; nitrogen \(= 4 \times 3 =\) 12 kg.
Level 3 · Further Application
Concrete is mixed as cement : sand : gravel = \(1:2:4\). A job needs 2.8 m³ of concrete.
1.6 m³
(a) Total parts = \(1 + 2 + 4 = 7\); one part = \(2.8 \div 7 = 0.4\) m³.
(b) Gravel = \(4 \times 0.4 =\) 1.6 m³.
A trail mix combines nuts : raisins : seeds in the ratio \(5:3:2\). A batch weighs 4 kg.
1.2 kg
(a) Total parts \(= 5 + 3 + 2 = 10\); one part \(= 4 \div 10 = 0.4\) kg.
(b) Raisins \(= 3 \times 0.4 =\) 1.2 kg.
In a lake study, 120 fish are tagged. Later a sample of 200 fish contains 15 tagged.
(a) 1600 fish (b) over-estimate
(a) \(N \approx \dfrac{120 \times 200}{15} = \dfrac{24\,000}{15} =\) 1600 fish.
(b) \(1600 > 1500\), so it is an over-estimate.
Level 1 · Fluency
Simplify the ratio \(18:24\).
\(3:4\)
Divide both by 6: \(3:4\).
Simplify the ratio \(45:30\).
\(3:2\)
Divide both by 15: \(3:2\).
Express 40 cm to 1 m as a ratio in simplest form.
\(2:5\)
Use the same units: 1 m \(=\) 100 cm, so \(40:100\); divide both by 20: \(2:5\).
Level 2 · Application
A profit of $4500 is shared between two partners in the ratio \(5:4\). Calculate each partner’s share.
$2000
Total parts = 9; one part = \(4500 \div 9 =\) $500. Shares: \(5 \times 500 =\) $2500 and \(4 \times 500 =\) $2000.
A sum of $840 is divided between three people in the ratio \(3:2:1\). Calculate each person’s share.
$420, $280, $140
Total parts \(= 6\); one part \(= 840 \div 6 = 140\). Shares: \(3 \times 140 =\) $420, \(2 \times 140 =\) $280, \(1 \times 140 =\) $140. Answer: $420, $280, $140.
A 3.5 kg mix of two nut types is made in the ratio \(3:4\). Find the mass of each type.
1.5 kg and 2 kg
Total parts \(= 7\); one part \(= 3.5 \div 7 = 0.5\) kg. Masses: \(3 \times 0.5 = 1.5\) kg and \(4 \times 0.5 = 2\) kg. Answer: 1.5 kg and 2 kg.
Level 3 · Further Application
A fruit punch mixes cordial : juice : soda in the ratio \(10:6:4\).
10 L
(a) 1 part = \(5 \div 10 = 0.5\) L; soda = \(4 \times 0.5 = 2\) L.
(b) Total parts = 20, so total = \(20 \times 0.5 =\) 10 L.
A drink mixes concentrate : water in the ratio \(2:7\). A jug holds 4.5 L of the drink.
1 L concentrate, 3.5 L water
Total parts \(= 9\); one part \(= 4.5 \div 9 = 0.5\) L.
(a) Concentrate \(= 2 \times 0.5 =\) 1 L.
(b) Water \(= 7 \times 0.5 =\) 3.5 L.
Three siblings share $1560 in the ratio \(5:4:3\).
$650, $520, $390; difference $260
(a) Total parts \(= 12\); one part \(= 1560 \div 12 = 130\). Shares: \(5 \times 130 =\) $650, \(4 \times 130 =\) $520, \(3 \times 130 =\) $390.
(b) \(650 - 390 =\) $260.
Level 1 · Fluency
On a map with scale \(1:1000\), 1 cm represents how many metres?
10 metres
1000 cm = 10 metres.
On a map with scale \(1:500\), 1 cm represents how many metres?
5 metres
\(1 \times 500 = 500\) cm = 5 metres.
On a map with scale \(1:2000\), 3 cm represents how many metres?
60 metres
\(3 \times 2000 = 6000\) cm = 60 metres.
Level 2 · Application
A map has a scale of \(1:25\,000\). Two towns are 6 cm apart on the map. Calculate the actual distance in kilometres.
1.5 km
\(6 \times 25\,000 = 150\,000\) cm = 1500 m = 1.5 km.
A map has a scale of \(1:50\,000\). Two features are 8 cm apart on the map. Calculate the actual distance in kilometres.
4 km
\(8 \times 50\,000 = 400\,000\) cm = 4000 m = 4 km.
On a map with scale \(1:150\,000\), how long (in cm) is a road that is actually 12 km long?
8 cm
12 km \(= 1\,200\,000\) cm; \(\div 150\,000 =\) 8 cm.
Level 3 · Further Application
A plan is drawn to a scale of \(1:3000\).
7 cm
(a) \(8 \times 3000 = 24\,000\) cm = 240 m.
(b) 210 m = \(21\,000\) cm; \(\div 3000 =\) 7 cm.
A map is drawn to a scale of \(1:40\,000\).
(a) 6 km (b) 15 cm
(a) \(15 \times 40\,000 = 600\,000\) cm = 6000 m = 6 km.
(b) 6 km \(= 600\,000\) cm; \(\div 40\,000 =\) 15 cm.
A plan uses a scale of \(1:250\).
(a) 17.5 m (b) 12 cm
(a) \(7 \times 250 = 1750\) cm = 17.5 m.
(b) 30 m \(= 3000\) cm; \(\div 250 =\) 12 cm.
Level 1 · Fluency
A room is 5 cm long on a \(1:100\) plan. What is its real length?
5 metres
\(5 \times 100 = 500\) cm = 5 metres.
A wall is 6 cm long on a \(1:50\) plan. What is its real length?
3 metres
\(6 \times 50 = 300\) cm = 3 metres.
A door is 2 cm wide on a \(1:40\) plan. What is its real width in metres?
0.8 metres
\(2 \times 40 = 80\) cm = 0.8 metres.
Level 2 · Application
On a \(1:200\) floor plan, a rectangular room measures 4 cm by 3 cm. Calculate its real area in square metres.
48 m²
Real dimensions: \(4 \times 200 = 800\) cm = 8 m, and \(3 \times 200 = 600\) cm = 6 m. Area = \(8 \times 6 =\) 48 m².
On a \(1:250\) plan, a rectangular courtyard measures 5 cm by 4 cm. Calculate its real area in square metres.
125 m²
Real dimensions: \(5 \times 250 = 1250\) cm = 12.5 m, and \(4 \times 250 = 1000\) cm = 10 m. Area \(= 12.5 \times 10 =\) 125 m².
On a \(1:100\) plan, a room measures 4.5 cm by 3 cm. Calculate its real perimeter in metres.
15 m
Real dimensions: \(4.5 \times 100 = 450\) cm = 4.5 m, and \(3 \times 100 = 300\) cm = 3 m. Perimeter \(= 2(4.5 + 3) =\) 15 m.
Level 3 · Further Application
A rectangular park is drawn 8 cm by 5 cm on a \(1:2500\) plan.
9.75 km/h
(a) \(8 \times 2500 = 20\,000\) cm = 200 m; \(5 \times 2500 = 12\,500\) cm = 125 m.
(b) Perimeter = \(2(200 + 125) = 650\) m = 0.65 km; time = 4 min = \(\dfrac{1}{15}\) h; speed = \(0.65 \div \dfrac{1}{15} =\) 9.75 km/h.
A rectangular garden is drawn 6 cm by 4 cm on a \(1:1500\) plan.
$64 800
(a) \(6 \times 1500 = 9000\) cm = 90 m; \(4 \times 1500 = 6000\) cm = 60 m.
(b) Area \(= 90 \times 60 = 5400\) m²; cost \(= 5400 \times 12 =\) $64 800.
A rectangular field is drawn 10 cm by 6 cm on a \(1:2000\) plan.
3.2 km
(a) \(10 \times 2000 = 20\,000\) cm = 200 m; \(6 \times 2000 = 12\,000\) cm = 120 m.
(b) Perimeter \(= 2(200 + 120) = 640\) m; 5 laps \(= 3200\) m = 3.2 km.
Level 1 · Fluency
On a building plan, what does the abbreviation “BED 1” usually indicate?
Bedroom 1.
On a building plan, what does the abbreviation “WC” usually indicate?
Water closet — that is, a toilet.
On a building plan, what does the abbreviation “KIT” usually stand for?
Kitchen.
Level 2 · Application
A plan marks a wall length as “3600”. Building plans use millimetres. State this length in metres.
3.6 m
3600 mm = 3.6 m.
A plan marks a room width as “2700” (in mm). State this length in metres.
2.7 m
2700 mm = 2.7 m.
A plan marks a hallway as “4500” (in mm), drawn to a scale of \(1:100\). State its real length in metres, and its length on the plan in centimetres.
4.5 m real; 4.5 cm on the plan
4500 mm = 4.5 m = 450 cm real; on a \(1:100\) plan, \(450 \div 100 =\) 4.5 cm.
Level 3 · Further Application
A plan shows a bathroom as \(2400 \times 1800\) (in mm).
4.32 m²
(a) 2400 mm = 2.4 m; 1800 mm = 1.8 m.
(b) Area = \(2.4 \times 1.8 =\) 4.32 m².
A plan shows a bedroom as \(3600 \times 3000\) (in mm).
10.8 m²
(a) 3600 mm = 3.6 m; 3000 mm = 3.0 m.
(b) Area \(= 3.6 \times 3.0 =\) 10.8 m².
A plan shows a kitchen as \(4200 \times 2500\) (in mm).
$472.50
(a) 4200 mm = 4.2 m; 2500 mm = 2.5 m; area \(= 4.2 \times 2.5 = 10.5\) m².
(b) Cost \(= 10.5 \times 45 =\) $472.50.
Level 1 · Fluency
The trapezoidal rule for one application is \(A \approx \dfrac{h}{2}(d_{L} + d_{R})\). What does \(h\) represent?
The width (perpendicular distance) between the two parallel measurements \(d_{L}\) and \(d_{R}\).
In the trapezoidal rule \(A \approx \dfrac{h}{2}(d_{L} + d_{R})\), what do \(d_{L}\) and \(d_{R}\) represent?
The two parallel offset lengths (the boundary measurements) at the left and right edges of the strip.
Use \(A \approx \dfrac{h}{2}(d_{L} + d_{R})\) with \(h = 10\) m, \(d_{L} = 8\) m and \(d_{R} = 12\) m to estimate the area.
100 m²
\(A \approx \dfrac{10}{2}(8 + 12) = 5 \times 20 =\) 100 m².
Level 2 · Application
A strip of land has parallel edges 25 m and 20 m apart by a perpendicular distance of 20 m. Use one application of the trapezoidal rule to estimate its area.
450 m²
\(A \approx \dfrac{20}{2}(25 + 20) = 10 \times 45 =\) 450 m².
A strip of land has parallel boundaries of 34 m and 28 m, a perpendicular distance of 15 m apart. Use one application of the trapezoidal rule to estimate its area.
465 m²
\(A \approx \dfrac{15}{2}(34 + 28) = 7.5 \times 62 =\) 465 m².
A field's two parallel sides measure 18 m and 24 m, and are 30 m apart. Use one application of the trapezoidal rule to estimate its area.
630 m²
\(A \approx \dfrac{30}{2}(18 + 24) = 15 \times 42 =\) 630 m².
Level 3 · Further Application
A block of land has three equally spaced offsets of 22 m, 30 m and 26 m, each 12 m apart, measured from a straight baseline.
648 m²
(a) \(A \approx \dfrac{12}{2}(22 + 2 \times 30 + 26) = 6 \times (22 + 60 + 26) = 6 \times 108 =\) 648 m².
(b) If the boundary bulges outward beyond the straight trapezium edges, the rule under-estimates and the true area is larger.
A tract of land has three offsets of 15 m, 21 m and 19 m, taken at equal intervals of 10 m along a baseline.
380 m²; $30 400
(a) \(A \approx \dfrac{10}{2}(15 + 2 \times 21 + 19) = 5 \times (15 + 42 + 19) = 5 \times 76 =\) 380 m².
(b) Price \(\approx 380 \times 80 =\) $30 400.
A river bank has offsets of 12 m, 18 m, 20 m and 14 m taken at equal 8 m intervals along a straight baseline.
408 m²
(a) \(A \approx \dfrac{8}{2}\big(12 + 14 + 2(18 + 20)\big) = 4 \times (26 + 76) = 4 \times 102 =\) 408 m².
(b) The first and last offsets are counted once; every middle offset is counted twice, all multiplied by \(\dfrac{h}{2}\).
Level 1 · Fluency
Using \(V = Ah\), find the volume of water on an area of 50 m² at a depth of 0.02 m.
1 m³
\(V = 50 \times 0.02 =\) 1 m³.
Using \(V = Ah\), find the volume of water on an area of 80 m² at a depth of 0.05 m.
4 m³
\(V = 80 \times 0.05 =\) 4 m³.
Using \(V = Ah\), find the volume of water on an area of 120 m² at a depth of 0.01 m.
1.2 m³
\(V = 120 \times 0.01 =\) 1.2 m³.
Level 2 · Application
25 mm of rain falls on a flat roof measuring 12 m by 8 m. Calculate the volume of water collected, in litres. (1 m³ = 1000 L.)
2400 litres
Depth = 25 mm = 0.025 m; area = 96 m²; \(V = 96 \times 0.025 = 2.4\) m³ = 2400 litres.
30 mm of rain falls on a flat roof measuring 10 m by 6 m. Calculate the volume of water collected, in litres. (1 m³ = 1000 L.)
1800 litres
Depth = 30 mm = 0.03 m; area = 60 m²; \(V = 60 \times 0.03 = 1.8\) m³ = 1800 litres.
18 mm of rain falls on a carpark measuring 20 m by 15 m. Calculate the volume of water collected, in litres. (1 m³ = 1000 L.)
5400 litres
Depth = 18 mm = 0.018 m; area = 300 m²; \(V = 300 \times 0.018 = 5.4\) m³ = 5400 litres.
Level 3 · Further Application
A rectangular roof is 15 m by 9 m. In a storm, 40 mm of rain falls.
(a) Area = 135 m²; depth = 0.04 m; \(V = 135 \times 0.04 = 5.4\) m³ = 5400 L.
(b) No — 5400 L exceeds the 5000 L tank, so it would overflow by 400 L.
A rectangular roof is 18 m by 11 m. In a storm, 35 mm of rain falls.
6930 L; overflow 930 L
(a) Area \(= 18 \times 11 = 198\) m²; depth = 35 mm = 0.035 m; \(V = 198 \times 0.035 = 6.93\) m³ = 6930 L.
(b) Overflow \(= 6930 - 6000 =\) 930 L.
A rectangular roof is 25 m by 12 m. During a storm, 22 mm of rain falls and all runoff is captured.
6600 L; 132 days
(a) Area \(= 25 \times 12 = 300\) m²; depth = 22 mm = 0.022 m; \(V = 300 \times 0.022 = 6.6\) m³ = 6600 L.
(b) \(6600 \div 50 =\) 132 days.