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Answer 1

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Given Radius and height of the Right Circular Cone are in ratio = 5 : 12

Given Volume of Cone = 314 cm^3

Let the Radius of The cone be 5x

and, Let the Height of the cone be 12x

Now, As we know that Volume of cone =

$= > \: \frac{1}{3} \pi \times {r}^{2} h \\ \\ \: therefore \: we \: can \: say \: the \: that \: \\ \\ \: \frac{1}{3} \pi \times {r}^{2} h \: = 314 \\ \\ \: \frac{1}{3} \times 3.14 \times (5x) {}^{2} \times (12x) = 314 \\ \\ \frac{1}{3} \times \frac{157}{50} \times {25x}^{2} \times 12x = 314 \\ \\ \frac{157}{50} \times {25x}^{2} \times 4x = 314 \\ \\ \frac{157}{25} \times {25x}^{2} \times 2x = 314 \\ \\ {157x}^{2} \times 2x = 314 \\ \\ {314x}^{3} = 314 \\ \\ {x}^{3} = 1 \\ \\ x = \sqrt[3]{1} \\ \\ x = 1$
Now x = 1

Then, Radius of cone = 5(1) = 5 cm

Height of cone = 12(1) = 12 cm

Total Surface Area of Cone = πr( L + r)

Now L(slant height) =

${l}^{2} = {r}^{2} + {h}^{2} \: \: (by \: pythagoras \: theorem) \\ \\ {l}^{2} = {5}^{2} + {12}^{2} \\ \\ {l}^{2} = 25 + 144 \\ \\ {l}^{2} = 169 \\ \\ l = \sqrt{169} \\ \\ l = 13cm$


Now, Slant height (L) = 13 cm

Therefore Total Surface Area of Cone =


$3.14 \times 5(13 + 5) \\ \\ 3.14 \times 5 \times 18 \\ \\ 282.6 \: cm {}^{ 2}$


Hope it would help you