(a) In the diagram, /PQ/ = 6 cm, /QR/ = 13 cm, /RS/ = 5 cm and < RSQ is a right- angled triangle. Calculate, correct to one decimal place, /PS/.
(b) The diagram show 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 7 cm. Calculate, correct to 3 significant figures, the surface area of the structure. [Take (pi = frac{22}{7})].
Explanation
(a)
(cos R = frac{5}{13} = 0.3846)
(R = cos^{-1} (0.3846) = 67.38°)
(/PS/^{2} = 5^{2} + 19^{2} – 2(5)(19)cos 67.38)
= (25 + 361 – 190(0.3872))
= (386 – 73.568)
(/PS/^{2} = 312.432)
(/PS/ = sqrt{312.432})
= (17.68 approxeq 17.7 cm)
(b) Slant height of a cone = (l = sqrt{7^{2} + 24^{2}})
(sqrt{49 + 576} = sqrt{625})
= (25 cm)
C.S.A of a cone = (pi rl)
= (frac{22}{7} times 7 times 25)
= (550 cm^{2})
Area of hemisphere = (frac{1}{2} times 4pi r^{2} = 2 pi r^{2})
(2 times frac{22}{7} times 7 times 7 = 308 cm^{2})
TSA of the structure = (550 cm^{2} + 308 cm^{2} = 858 cm^{2})