Higher June 2018 Paper 1R Q22
22 A 3-D shape consists of a hollow sphere made of metal.
The diagram shows a cross section drawn through the centre, \(O\), of the sphere.
Diagram NOT accurately drawn
The internal radius of the sphere is 1.2 m.
The thickness of the metal is \(t\) cm.
The density of the metal is 2700 kg per m3
The mass of the 3-D shape is 1980 kg.
Work out the value of \(t\).
Give your answer correct to 2 significant figures.
(5)
| Scheme | Marks |
|---|---|
\(\dfrac{4}{3} \times \pi \times R^3 - \dfrac{4}{3} \times \pi \times 1.2^3\) or \(\dfrac{4}{3} \times \pi \times (1.2 + t)^3 - \dfrac{4}{3} \times \pi \times 1.2^3\) | M1 |
| \(\left(\dfrac{4\pi}{3}R^3 - \dfrac{4}{3} \times \pi \times 1.2^3\right) \times 2700 = 1980\) | M1 |
\(\dfrac{4\pi}{3}R^3 = \dfrac{4}{3} \times \pi \times 1.2^3 + \dfrac{1980}{2700}\) = 7.238229474 + 0.7333333 = 7.97(1562807) | M1 |
\(R = \sqrt[3]{\dfrac{3}{4\pi} \times \left(\dfrac{4}{3} \times \pi \times 1.2^3 + \dfrac{1980}{2700}\right)}\) = 1.2392… 1.2392 − 1.2 = 0.0392 | M1 |
| 3.9 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for an expression for the volume of the inner sphere
M1: for a correct expression
or sight of 7.23(8229474) + 0.73(33333)
or sight of 7.97(1562807)
M1: for a correct expression
or sight of \(\sqrt[3]{1.90(3070437)}\)
or sight of 1.23(9229151)
or sight of 0.0392(29151)
A1: for 3.9 – 3.92