Question

29. The radius and height of a right circular cone are in the ratio 4:3 and its volume is 2156 cu cm
Find the curved surface area and the total surface area of the cone.​

Answer 1

EXPLANATION.

=> The radius and height of a right circular cone

are in the ratio = 4:3.

=> it's volume = 2156 cu cm³

TO FIND C.S.A AND T.S.A OF CONE.

$\sf : \implies \: let \: the \: radius \: of \: cone \: = 4x \\ \\ \sf : \implies \: let \: the \: height \: of \: cone \: = 3x \\ \\ \sf : \implies \: volume \: = 2156 \: cu \: cm {}^{3}$

$\sf : \implies \green{{\: volume \: of \: cone \: = \frac{1}{3}\pi \: {r}^{2} h}}$

$\sf : \implies \: \dfrac{1}{3} \times \pi \times \: (4x) {}^{2} \times 3x = 2156 \\ \\ \sf : \implies \: \frac{1}{ \cancel{3}} \times \frac{22}{7} \times 16 {x}^{2} \times \cancel{3}x = 2156 \\ \\ \sf : \implies \: \cancel{22} \times 16 {x}^{2} \times x = \cancel{2156} {}^{98} \times 7 \\ \\ \sf : \implies \: 16 {x}^{3} = 98 \times 7 \\ \\ \sf : \implies \: x = \frac{7}{2} = 3.5 \: cm$

$\sf : \implies \: radius \: of \: cone \: = 4x = 4 \times 3.5 = 14 \: cm \\ \\ \sf : \implies \: \: height \: of \: cone \: = 3x = 3 \times 3.5 = 10.5 \: cm$

$\sf : \implies \: slant \: height \: = l \: = \sqrt{ {r}^{2} + {h}^{2} } \\ \\ \sf : \implies \: l = \sqrt{ {14}^{2} + {10.5}^{2} } \\ \\ \sf : \implies \: l = \sqrt{306.25} = 17.5 \: cm$

$\sf : \implies \: curved \: surface \: area \: of \: cone = \pi \: rl \\ \\ \sf : \implies \: \frac{22}{7} \times 14 \times 10.5 \\ \\ \sf : \implies \: 44 \times 17.5 = \: 770 \: cm {}^{2}$

Answer 2

Step-by-step explanation:

Let the radius be 4x and height be 3x of the right circular cone.

We have,

• Area of the right circular cone = 2156 cm³

★ According to Question now,

➨ Volume of cone = ⅓ πr²h

➳ 2156 = 1/3 × 22/7 × (4x)² × (3x)

➳ 2156 = 1/3 × 22/7 × 16x² × 3x

➳ 2156 = 22/7 × 16x² × x

➳ 16x³ = 2156 × 7/22

➳ x³ = 98 × 7/16

➳ x³ = 7 × 7 × 7/2 × 2 × 2

➳ x = 7/2

➳ x = 3.5 cm

Therefore,

• Radius = 4x = 4(3.5) = 14 cm

• Height = 3x = 3(3.5) = 10.5 cm

______________________

➳ Slant height (l)² = r² + h²

➳ l² = (14)² + (3.5)²

➳ l² = 196 + 110.25

➳ l² = 306.25

➳ l = √306.25

➳ Slant Height (l) = 17.5 cm

_____________________

➳ CSA of cone = πrl

➳ CSA of cone = 22/7 × 14 × 10.5

➳ CSA of cone = 22 × 2 × 10.5

➳ CSA of cone = 44 × 10.5

➳ CSA of cone = 770 cm²

___________________________

➳ TSA of cone = πr² + πrl

➳ TSA of cone = 22/7 × 14² + 770

➳ TSA of cone = 616 + 770

➳ TSA of cone = 1386 cm²