Question

The radius and height of a solid right-circular cone are in the ratio of 5 : 12. If its volume is 314 cm³, find its total surface area. [Take π = 3.14.]

Answer 1

Given: The radius : height = 5 : 12. Volume = 314 cm^3. Take π = 3.14.

Step-wise calculation:

1. Let radius = 5k and height = 12k.

2. Volume of cone = `(1/3) πr^2h` 

= `(1/3) π (5k)^2 (12k)`

= `(1/3) π xx 25k^2 xx 12k`

= 100 πk³

3. Set 100 πk³ = 314. 

With π = 3.14:

100 × 3.14 × k³ = 314 

⇒ 314k³ = 314 

⇒ k³ = 1

⇒ k = 1

4. So r = 5 × 1 = 5 cm, h = 12 × 1 = 12 cm.

5. Slant height `l = sqrt(r^2 + h^2)` 

= `sqrt(5^2 + 12^2)` 

= `sqrt(169)`

= 13 cm

6. Total surface area (TSA) of cone = πr² + πrl 

= πr(r + l) 

= π × 5 × (5 + 13)

= π × 5 × 18 

= 90π

7. With π = 3.14:

TSA = 90 × 3.14 

= 282.6 cm²

The total surface area of the cone is 282.6 cm².