(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 = \(\frac{1}{2} \times 4\pi r^{2} = 2 \pi r^{2}\)
\(2 \times \frac{22}{7} \times 7 \times 7 = 308 cm^{2}\)
TSA of the structure = \(550 cm^{2} + 308 cm^{2} = 858 cm^{2}\)