Theory question 110
Mensuration of Solids(a) The total surface area of two spheres are in the ratio 9:49. If the radius of the smaller sphere is 12 cm, find, correct to the nearest cm3, the volume of the bigger sphere.
(b) A cyclist starts from a point X and rides 3 km due West to a point Y. At Y, he changes direction and rides 5 km North-West to a point Z. (i) How far is he from the starting point, correct to the nearest km? (ii) Find the bearing of Z from X, to the nearest degree.
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(a) r2 : R2 = 9 : 49, so r : R = 3 : 7 and R = 12 × 7/3 = 28 cm.
Volume = 4/3 × 22/7 × 283 ≈ 91,989 cm3.
(b)(i) ∠XYZ = 90° + 45° = 135°. XZ2 = 32 + 52 − 2(3)(5) cos 135° = 34 + 21.21 = 55.21, so XZ ≈ 7.43 ≈ 7 km.
(ii) sin ∠YXZ = 5 sin 135°/7.43 ≈ 0.476, so ∠YXZ ≈ 28.4°. Bearing of Z from X = 270° + 28.4° ≈ 298°.
Ratio of surface areas = (ratio of radii)2.