Correct Answer - Option 2 : 28 cm
Given:
Total surface area : Curved surface area = 8 : 3
Volume of cone = 12936 cm3
Formula used:
Total surface area (T.S.A) = πr(r + l)
Curved surface area (C.S.A) = πrl
\({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{l}}^{2{\rm{\;}}}} - {\rm{\;}}{{\rm{r}}^2}} \)
Volume = \(\left( {\frac{1}{3}} \right) \times {\rm{\pi }} \times {{\rm{r}}^2} \times {\rm{h}}\)
l = slant height of a cone
r = radius of the base
h = height of a cone
Calculation:
πr(r + l) : πrl = 8x : 5x
⇒ (r + l) : l = 8x : 5x
⇒ (r + l – l) = (8x – 5x)
⇒ r = 3x
\({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{l}}^{2{\rm{\;}}}} - {\rm{\;}}{{\rm{r}}^2}} \)
⇒ \({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{(5x)}}^{2{\rm{\;}}}} - {\rm{\;}}{{\rm{(3x)}}^2}} \)
⇒ \({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{25x}}^{2{\rm{\;}}}} - {\rm{\;}}{{\rm{9x}}^2}} \)
⇒ \({\rm{h}} = {\rm{\;}}\sqrt {16{{\rm{x}}^2}} \)
⇒ h = 4x
\(\left( {\frac{1}{3}} \right) \times {\rm{\pi }} \times {{\rm{r}}^2} \times {\rm{h}}\) = 12936
⇒\(\left( {\frac{1}{3}} \right) \times {\frac{22}{7}} \times {{\rm{9x}}^2} \times {\rm{4x}}\) = 12936
⇒\(\frac{{264{{\rm{x}}^3}}}{7}{\rm{\;}}\)= 12936
⇒ \(\frac{{{{\rm{x}}^3}}}{7}{\rm{\;}}\)= 49
⇒ x3 = 343
⇒ x = 7
⇒ 4x = 28
∴ Height of the cone is 28 cm
