A toy in the form of a cone of radius 3.5cm mounted on a n hemisphere of same radius. The total height of the toy is 15.5cm. Find the total surface area of the solid.
Answers
Given :
- Radius of cone = 3.5 cm
- Height of cone = 15.5 cm
- Radius of hemisphere = 3.5 cm
To Find :
- The total surface area of solid = ?
Solution :
To Calculate Lateral Surface Area of cone first of all we will calculate the Slant Height of the cone :
- Radius of cone = 3.5 cm
- Height of cone = Total height of toy - radius of hemisphere = 15.5 - 3.5 = 12
- Slant Height of cone = ?
Let's calculate the Slant Height of cone :
→ Slant Height = √(Radius)² + (Height)²
→ Slant Height = √(3.5)² + (12)²
→ Slant Height = √12.25 + 144
→ Slant Height = √156.25
→ Slant Height = 12.5 cm
- Hence, slant height of the cone is 12.5 cm.
Now, let's calculate CSA of hemisphere :
➻ CSA of hemisphere = 2πr²
➻ CSA of hemisphere = 2 × 22/7 × (3.5)²
➻ CSA of hemisphere = 2 × 22/7 × 12.25
➻ CSA of hemisphere = 2 × 22 × 1.75
➻ CSA of hemisphere = 44 × 1.75
➻ CSA of hemisphere = 77 cm²
- Hence, Lateral surface area of hemisphere is 77 cm².
Now, let's calculate the CSA of cone :
➺ CSA of cone = πrl
➺ CSA of cone = 22/7 × 3.5 × 12.5
➺ CSA of cone = 22 × 3.5 × 1.75
➺ CSA of cone = 137.5 cm²
- Hence, Lateral surface area of cone is 137.5 cm².
Now, we can calculate the TSA of the solid :
⊷ TSA of solid = CSA of cone + CSA of hemisphere
⊷ TSA of solid = 137.5 + 77
⊷ TSA of solid = 214.5 cm²
- Hence, the total surface area of solid is 214.5 cm².
☆Answer☆
Radius of cone = 3.5 cm
Radius of hemisphere = 3.5 cm
Height of the toy = 15.5 cm
Now,
Height of cone = Height of toy - Radius of hemisphere
= (15.5-3.5) cm
= 12 cm
Slant height of the cone,l = _/R^2+H^2
= _/(3.5)^2+(12)^2
= _/12.25+144
= 12.5 cm
CSA of cone = πrl
= (22/7)×3.5x12.5
= (22×3.5×1.75) cm^2
= 134.75 cm^2
CSA of hemisphere = 2πr^2
= 2×(22/7)×(3.5)^2
= 44×1.75
= 77 cm^2
Now, TSA of the toy
= CSA of cone + CSA of hemisphere
= (134.75+77) cm^2
= 211.75 cm^2
Hence, the total surface area of toy is 211.75 cm^2.
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