The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height and the volume of the cone.
Answer: 4π(5)2 = 5 × π(4)l ⇒ l = 5 cm; h = √(25 − 16) = 3 cm; volume = 13π × 16 × 3 = 16π ≈ 50.29 cm3.
Step-by-step solution
Given: Sphere radius 5 cm; Cone radius 4 cm; SA of sphere = 5 × CSA of cone
To find: Height and volume of the cone
To find: Height and volume of the cone
Idea: Use the given relation to find the cone’s slant height, then Pythagoras for the height, then the volume formula.
- Surface area of sphere = 4π × 52 = 100π cm2.½ mark
- 100π = 5 × π × 4 × l = 20πl ⇒ l = 5 cm.1 mark
- h = √(l2 − r2) = √(25 − 16) = √9 = 3 cm.1 mark
- Volume = 13πr2h = 13 × π × 16 × 3 = 16π ≈ 16 × 227 ≈ 50.29 cm3.1½ marks
Height = 3 cm; volume = 16π ≈ 50.29 cm³ (π = 22/7; 50.24 cm³ with π = 3.14).
Check: CSA of cone = π × 4 × 5 = 20π; 5 × 20π = 100π = 4π × 25 ✓.
Answer to write in the exam
4π(5)2 = 5 × π(4)l
100π = 20πl ⇒ l = 5 cm
h = √(52 − 42) = 3 cm
V = 13π × 42 × 3 = 16π
∴ h = 3 cm, V = 16π ≈ 50.29 cm3
Common mistakes that cost marks
- Taking l as the height and finding volume with h = 5.
- Putting the factor 5 on the wrong side (sphere = 5 × cone, not cone = 5 × sphere).
How this can come in the exam
MCQ (1 mark)
A cone has radius 6 cm and curved surface area 60π cm2. Its height is
- 10 cm
- 8 cm
- 6 cm
- 4 cm
Show answer
(B) 8 cm
6l = 60 ⇒ l = 10; h = √(100 − 36) = 8 cm.
Try one yourself
The surface area of a sphere of radius 10 cm is five times the curved surface area of a cone of radius 8 cm. Find the height and the volume of the cone.
Show answer
400π = 5 × π × 8 × l ⇒ l = 10 cm; h = √(100 − 64) = 6 cm; V = 13π × 64 × 6 = 128π cm3.
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