Rain falling on a shed roof runs into a 150 litre water butt. The roof is the rectangle shown, each length correct to 1 decimal place. One storm drops 18 mm...
Question 1 Report
Rain falling on a shed roof runs into a 150 litre water butt. The roof is the rectangle shown, each length correct to 1 decimal place. One storm drops 18 mm of rain, to the nearest millimetre.
- Calculate the upper bound of the roof area. (2)
- Calculate the lower bound of the roof area. (1)
- Using upper bounds, find the greatest volume of rain landing on the roof, in litres. (2)
- Explain whether the butt could overflow. (1)
Answer Details
The figure shows a rectangular roof measuring 3.2 m by 2.5 m. Each length is to 1 decimal place, so the true length lies between 3.15 m and 3.25 m and the true width between 2.45 m and 2.55 m.
(a) Area is a product, so it is greatest when both sides take their upper bounds.
- Upper bound \(= 3.25 \times 2.55\). [1]
- \(= 8.2875\) m\(^2\). [1]
(b) Lower bound \(= 3.15 \times 2.45 = 7.7175\) m\(^2\). [1]
(c) The rain forms a thin layer over the roof, so the volume is area times depth, with the depth converted to metres.
- The greatest rainfall is \(18.5\) mm \(= 0.0185\) m, so the greatest volume is \(8.2875 \times 0.0185 = 0.15331875\) m\(^3\). [1]
- Since 1 m\(^3\) is 1000 litres, this is \(0.15331875 \times 1000 = 153.3\) litres to 1 decimal place. [1]
(d) The butt holds 150 litres, but as much as about 153 litres could arrive, and \(153 > 150\), so the butt could overflow. [1]
The rainfall of 18 mm to the nearest millimetre has a half-unit of 0.5 mm, giving an upper bound of 18.5 mm. Converting millimetres to metres by dividing by 1000, rather than by 100, is essential: using 0.185 m would give a volume ten times too large and a completely different conclusion.
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