Higher June 2018 Paper 1 Q18
18 Here is a tunnel for a toy train.
The diagram below shows the cross section of the tunnel.
Not drawn accurately
\(AD\) is a semicircular arc of radius 10 cm
\(BC\) is a semicircular arc of radius 7 cm
The length of the tunnel is 30 cm
Work out the total area of all six faces of the tunnel.
Give your answer in terms of \(\pi\). [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\pi \times 10^2 - \pi \times 7^2\) or \(100\pi - 49\pi\) or \(51\pi\) or \(\dfrac{1}{2} \times \pi \times 10^2 - \dfrac{1}{2} \times \pi \times 7^2\) or \(\dfrac{1}{2} \times 100\pi - \dfrac{1}{2} \times 49\pi\) or \(\dfrac{1}{2} \times 51\pi\) or \(25.5\pi\) | M1 | oe implied by \(102\pi\) method to work out front and/or back faces – must not be part of a method to work out volume (\(\times\) 30) may be taken to be full circles |
| \(2 \times \pi \times 10 \times 30\) or \(600\pi\) or \(\dfrac{1}{2} \times 2 \times \pi \times 10 \times 30\) or \(300\pi\) or \(2 \times \pi \times 7 \times 30\) or \(420\pi\) or \(\dfrac{1}{2} \times 2 \times \pi \times 7 \times 30\) or \(210\pi\) or \(1020\pi\) or \(510\pi\) | M1 | oe method to work out outer and/or inner curved surfaces may be taken to be full circles \(1122\pi\) implies M1M1 |
| \(\left(\dfrac{1}{2} \times \pi \times 10^2 - \dfrac{1}{2} \times \pi \times 7^2\right) \times 2\) \(+ \dfrac{1}{2} \times 2 \times \pi \times 10 \times 30\) \(+ \dfrac{1}{2} \times 2 \times \pi \times 7 \times 30\) or \(2 \times 25.5\pi + 300\pi + 210\pi\) or \(561\pi\) | M1dep | oe dep on M1M1 correct method to work out total of front, back, outer curved and inner curved surfaces |
| \(2 \times 30 \times 3\) or 180 | M1 | implied by an answer of \(n\pi + 180\) do not award if 180 is used as \(180\pi\) |
| \(561\pi + 180\) | A1 |
Additional guidance
| \(150\pi\) and \(105\pi\) implies use of radius for curved surface areas | max M1M0M0M1A0 |
| Condone use of [3.14, 3.142] for \(\pi\) up to M1M1M0M1A0 |