Step-by-step explanation:
Let common multiplier of the given ratios be x
Therefore,
Slant height l = 5x
Perpendicular height h = 4x
& radius r = 6 cm (given)
[tex] \because {l}^{2} = {h}^{2} + {r}^{2} \\ \therefore \: {(5x)}^{2} = {(4x)}^{2} + {6}^{2} \\ \therefore \: 25 {x}^{2} = 16 {x}^{2} + 36 \\ \therefore \: 25 {x}^{2} - 16 {x}^{2} = 36 \\ \therefore \: 9 {x}^{2} = 36 \\ \therefore \: {x}^{2} = \frac{36}{9} \\ \therefore \: {x}^{2} = 4 \\ \therefore \: {x} = 2 \\ \\ \implies \: 5x = 5 \times 2 = 10 \\ \therefore \:slant \: height \: = 10 \:cm. \\ \\ curved \: surface \: area \: of \: cone \\ = \pi \: r \: l \\ = 3.14 \times 6 \times 10 \\ = 188.4 \: {cm}^{2} \\ \\ total \: \: surface \: area \: of \: cone \\ = \pi \: r(l + r) \\ = 3.14 \times 6(10 + 6) \\ = 18.84 \times 16 \\ = 301.44 \: {cm}^{2} [/tex]
