The height of a right circular cone is 21 cm and area of its curved surface area is 3 times the area of its base, then what is the volume (Approx.) of the cone?
1. 1213 cm3
2. 1212 cm3
3. 1214 cm3
4. 1215 cm3
The height of a right circular cone is 21 cm and area of its curved surface area is 3 times the area of its base, then what is the volume (Approx.) of the cone?
1. 1213 cm3
2. 1212 cm3
3. 1214 cm3
4. 1215 cm3
1. 1213 cm3
2. 1212 cm3
3. 1214 cm3
4. 1215 cm3
Correct Answer – Option 1 : 1213 cm3
Given:
Height of the cone = 21 cm
Formula used:
Curved surface area of a right circular cone = πrl
Volume of a cone = \(\left( {\frac{1}{3}} \right) \times {\rm{\pi }} \times {{\rm{r}}^2} \times {\rm{h}}\)
Area of circle = πr2
\({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{l}}^{2{\rm{\;}}}} – {\rm{\;}}{{\rm{r}}^2}} \)
Where r = radius, l = slant height, h = height of a cone
Calculation:
According to the question,
πrl = 3πr2
⇒ l = 3r
⇒ l/r = 3/1
\({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{l}}^{2{\rm{\;}}}} – {\rm{\;}}{{\rm{r}}^2}} \)
⇒ \({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{3}}^{2{\rm{\;}}}} – {\rm{\;}}{{\rm{1}}^2}} \)
⇒ \({\rm{h}} = {\rm{\;}}\sqrt {{{\rm{9}{\rm{\;}}}} – {\rm{\;}}{{\rm{1}}}} \)
⇒ h = √8
⇒ h = 2√2
⇒ 21 = 2√2
⇒ r = 21/2√2
\(\left( {\frac{1}{3}} \right) \times {\rm{\pi }} \times {{\rm{r}}^2} \times {\rm{h}}\) = \(\left( {\frac{1}{3}} \right) \times {\frac{22}{7}} \times ({{\frac{21}{2\sqrt 2}})^2} \times {\rm{21}}\)
⇒\(\left( {\frac{1}{3}} \right) \times {\frac{22}{7}} \times ({{\frac{441}{8}})} \times {\rm{21}}\)
⇒ 22 × 63/8 × 7
⇒ 1212.75
⇒ Volume = 1212.75 cm3 ≈ 1213 cm3
∴ Volume of the cone is 1213 cm3