Theory question 113
Trigonometry(a) In the diagram, |PQ| = 6 cm, |QR| = 13 cm, |RS| = 5 cm and ∠RSQ is a right angle. Calculate, correct to one decimal place, |PS|.
(b) The diagram shows a wooden structure in the form of a cone mounted on a hemispherical base. The vertical height of the cone is 24 cm and the base radius 7cm. Calculate, correct to 3 significant figures, the surface area of the structure. [Take π = 22/7]
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(a) |SQ| = √(132 − 52) = 12 cm. cos ∠SQR = 12/13. P, Q, R are on a straight line, so cos ∠PQS = −12/13.
|PS|2 = 62 + 122 − 2(6)(12)(−12/13) = 180 + 132.9 = 312.9, so |PS| ≈ 17.7 cm.
(b) Slant height = √(242 + 72) = 25 cm.
Cone curved surface = 22/7 × 7 × 25 = 550 cm2; hemisphere curved surface = 2 × 22/7 × 72 = 308 cm2.
Surface area = 858 cm2.
Only the curved surfaces are on the outside; the flat circle is hidden where the two parts join.