Mathematics Standard • Year 12 • Ratios and rates • Lesson 5
Capture-Recapture and Energy, Skill Drill
Build fluency in the two mechanics of this lesson: substituting into the capture-recapture formula, and moving from a watt rating to kilowatt hours and then to dollars.
1. Quick recall
Answer in the space provided. 1 mark each
Q1.1 Write the capture-recapture formula for the population N.
Q1.2 To convert watts into kilowatts you divide by ____________.
Q1.3 Energy in kilowatt hours equals power in kilowatts multiplied by ____________.
2. Worked example, capture-recapture
Follow each line. Every step has its reason.
Problem. 40 fish are tagged. Later 50 are caught and 8 are tagged. Estimate the population.
Step 1, name the three numbers.
n1 = 40 marked, n2 = 50 caught, m2 = 8 of those marked
Reason: mixing up n2 and m2 is the most common error, so label them before substituting.
Step 2, set the two fractions equal.
8/50 = 40/N
Reason: the sample is 16% marked, so the whole lake should be about 16% marked.
Step 3, solve and check.
N = (40 × 50) ÷ 8 = 250 fish, and 40 ÷ 250 = 0.16 = 8 ÷ 50
Reason: checking that both fractions match catches a flipped formula instantly.
3. Capture-recapture
Estimate each population. 1 mark each
Q3.1 25 marked, 40 caught later, 5 of them marked.
Q3.2 60 marked, 75 caught later, 15 of them marked.
Q3.3 100 marked, 120 caught later, 24 of them marked.
4. Watts to kilowatts
Convert each power rating. 1 mark each
Q4.1 2400 W
Q4.2 750 W
Q4.3 5000 W
5. Energy and cost
Convert to kilowatts first, then multiply by the hours, then by the price. 2 marks each
Q5.1 A 2 kW heater runs for 4 hours. Find the energy used, in kilowatt hours.
Q5.2 A 1200 W kettle runs for a total of 3 hours over a week. Find the energy used.
Q5.3 Electricity costs $0.30 per kilowatt hour. Find the cost of the 8 kWh used by an oven.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, The formula
N = (n1 × n2) ÷ m2, where n1 were marked, n2 were caught later and m2 of those carried a mark.
Q1.2, Watts to kilowatts
Divide by 1000.
Q1.3, Energy
The time in hours. Power does not include time; energy does.
Q3.1, 25 marked, 5 of 40
N = (25 × 40) ÷ 5 = 1000 ÷ 5 = 200. Check: 5 ÷ 40 = 0.125 and 25 ÷ 200 = 0.125.
Q3.2, 60 marked, 15 of 75
N = (60 × 75) ÷ 15 = 4500 ÷ 15 = 300. Check: 15 ÷ 75 = 0.2 and 60 ÷ 300 = 0.2.
Q3.3, 100 marked, 24 of 120
N = (100 × 120) ÷ 24 = 12 000 ÷ 24 = 500. Check: 24 ÷ 120 = 0.2 and 100 ÷ 500 = 0.2.
Q4.1, 2400 W
2400 ÷ 1000 = 2.4 kW.
Q4.2, 750 W
750 ÷ 1000 = 0.75 kW. A power under one kilowatt is perfectly normal for a small appliance.
Q4.3, 5000 W
5000 ÷ 1000 = 5 kW.
Q5.1, 2 kW for 4 hours
2 × 4 = 8 kWh.
Q5.2, 1200 W kettle for 3 hours
Convert first: 1200 W = 1.2 kW. Then 1.2 × 3 = 3.6 kWh. Multiplying 1200 by 3 before converting gives 3600 watt hours, which is the same energy but a thousand times the number, and it will wreck the cost line.
Q5.3, 8 kWh at $0.30
8 × 0.30 = $2.40. Check: 2.40 ÷ 0.30 = 8 kWh.