Mathematics Standard • Year 12 • Ratios and rates • Lesson 4
Irregular Blocks and Rainfall, Application
Work out the strip width from a total length, estimate the area of an irregular block, then turn that area into a volume of water. This is the shape of the HSC question.
1. Find h first
Each block is measured with three equally spaced offsets. The question gives the total length, not the strip width. 2 marks each
Q1.1 A block is 30 m long. The offsets are 14 m, 20 m and 16 m. Estimate the area.
Q1.2 A block is 16 m long. The offsets are 9 m, 13 m and 11 m. Estimate the area.
2. Rain on those blocks
Use your areas from section 1. 3 marks each
Q2.1 22 mm of rain falls on the block from Q1.1. Find the volume of water, in litres.
Q2.2 35 mm of rain falls on the block from Q1.2. Find the volume of water, in litres.
3. A roof
No trapezoidal rule needed here, just the rainfall chain. 3 marks
Q3.1 A rectangular roof measures 14 m by 7 m. Find the volume of water, in litres, from 30 mm of rain.
4. Working backwards
Both parts use the same 60 square metre roof. 2 marks each
Q4.1 How much rain, in millimetres, is needed to collect 3000 L from a 60 square metre roof?
Q4.2 How much water, in litres, does the same roof collect from 12 mm of rain?
5. The full chain
Set it out in order: h, area, depth in metres, volume, litres. 4 marks
Q5.1 A paddock is 36 m long with three equally spaced offsets of 22 m, 30 m and 26 m. After a storm the rain gauge reads 25 mm. Estimate the area of the paddock and the volume of water that fell on it, in litres.
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What I'll revisit before next class:
Q1.1, Block 30 m long
Three offsets make two strips, so h = 30 ÷ 2 = 15 m. A = (15/2)(14 + 2 × 20 + 16) = 7.5 × 70 = 525 square metres. Check as two strips: 7.5(14+20) = 255 and 7.5(20+16) = 270, and 255 + 270 = 525.
Q1.2, Block 16 m long
h = 16 ÷ 2 = 8 m. A = (8/2)(9 + 2 × 13 + 11) = 4 × 46 = 184 square metres.
Q2.1, 22 mm on 525 square metres
Depth = 0.022 m. Volume = 525 × 0.022 = 11.55 cubic metres = 11 550 L. Check: 11.55 ÷ 525 = 0.022.
Q2.2, 35 mm on 184 square metres
Depth = 0.035 m. Volume = 184 × 0.035 = 6.44 cubic metres = 6440 L.
Q3.1, Roof 14 m by 7 m, 30 mm
Area = 98 square metres. Depth = 0.03 m. Volume = 98 × 0.03 = 2.94 cubic metres = 2940 L.
Q4.1, Rain needed for 3000 L
3000 L = 3 cubic metres. Since V = A × d, the depth is d = 3 ÷ 60 = 0.05 m, which is 50 mm. Check: 60 × 0.05 = 3 cubic metres = 3000 L.
Q4.2, 12 mm on the same roof
Volume = 60 × 0.012 = 0.72 cubic metres = 720 L.
Q5.1, The full chain
h = 36 ÷ 2 = 18 m [1]. A = (18/2)(22 + 2 × 30 + 26) = 9 × 108 = 972 square metres [1]. Depth = 25 mm = 0.025 m [1]. Volume = 972 × 0.025 = 24.3 cubic metres = 24 300 L [1]. Check: 24.3 ÷ 972 = 0.025. The area is an estimate, so the volume is too.