Mathematics Standard • Year 12 • Ratios and rates • Lesson 4
Trapezoidal Rule, Skill Drill
Build fluency in substituting into the trapezoidal rule for one and two applications, and in turning a rainfall depth into a volume of water.
1. Quick recall
Answer in the space provided. 1 mark each
Q1.1 Write the trapezoidal rule for one application.
Q1.2 In the rule, what does h stand for?
Q1.3 To change a rainfall depth from millimetres into metres you divide by ____________.
2. Worked example, one application
Follow each line. Every step has its reason.
Problem. Two offsets of 12 m and 16 m are 8 m apart. Estimate the area.
Step 1, write the rule and name the parts.
A = (h/2)(df + dl), with h = 8, df = 12, dl = 16
Reason: the rule is on the reference sheet, so the marks are in picking the right numbers.
Step 2, halve the strip width first.
A = (8/2)(12 + 16) = 4 × 28
Reason: the bracket is the SUM of the offsets. The half turns it into the average.
Step 3, finish and check.
A = 112 square metres, and 112 ÷ 4 = 28 = 12 + 16
Reason: it is an estimate, so write "approximately" in your answer.
3. One application
Estimate each area. 1 mark each
Q3.1 Offsets of 7 m and 9 m, which are 4 m apart.
Q3.2 Offsets of 5 m and 11 m, which are 10 m apart.
Q3.3 Offsets of 8 m and 10 m, which are 6 m apart.
4. Two applications
Three offsets, so the middle one is doubled. 2 marks each
Q4.1 Offsets of 5 m, 8 m and 6 m, each 4 m apart.
Q4.2 Offsets of 12 m, 18 m and 14 m, each 10 m apart.
5. Volume of rainfall
Convert the depth to metres before you multiply. 2 marks each
Q5.1 Write a rainfall of 10 mm in metres.
Q5.2 A flat roof measures 8 m by 5 m. Find the volume of water, in litres, from 15 mm of rain.
Q5.3 A paddock has an area of 500 square metres. Find the volume of water, in litres, from 20 mm of rain.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1, The rule
A = (h/2)(df + dl), where df is the first offset and dl is the last.
Q1.2, What h is
The strip width, the distance along the baseline between the two offsets. The strips must be equally spaced.
Q1.3, Millimetres to metres
Divide by 1000. So 25 mm becomes 0.025 m.
Q3.1, Offsets 7 and 9, h = 4
A = (4/2)(7 + 9) = 2 × 16 = 32 square metres.
Q3.2, Offsets 5 and 11, h = 10
A = (10/2)(5 + 11) = 5 × 16 = 80 square metres.
Q3.3, Offsets 8 and 10, h = 6
A = (6/2)(8 + 10) = 3 × 18 = 54 square metres.
Q4.1, Offsets 5, 8, 6 with h = 4
A = (4/2)(5 + 2 × 8 + 6) = 2 × 27 = 54 square metres. Check as two strips: 2(5+8) = 26 and 2(8+6) = 28, and 26 + 28 = 54.
Q4.2, Offsets 12, 18, 14 with h = 10
A = (10/2)(12 + 2 × 18 + 14) = 5 × 62 = 310 square metres. Only the middle offset is doubled.
Q5.1, 10 mm in metres
10 ÷ 1000 = 0.01 m.
Q5.2, Roof 8 m by 5 m, 15 mm of rain
Area = 40 square metres. Depth = 0.015 m. Volume = 40 × 0.015 = 0.6 cubic metres = 600 L. Check: 0.6 ÷ 40 = 0.015.
Q5.3, Paddock 500 square metres, 20 mm of rain
Depth = 0.02 m. Volume = 500 × 0.02 = 10 cubic metres = 10 000 L. Ten cubic metres of water weighs about ten tonnes.