If the ratio of the height to the slant height of a cone is 4 : 5 and its volume is 12936 cm³ , then what will be its diameter? (Use value of π as \(\frac{22}{7}\))

Given:

Ratio of height (h) to slant height (l) = 4 : 5

Volume of cone (V) = 12936 cm³

Formula Used:

l² = r² + h² (where r is radius)

Volume of cone (V) = \(\frac{1}{3}\)\(\pi\)r²h

Diameter = 2r

Calculation:

Let h = 4x and l = 5x

Using Pythagoras theorem: l² = r² + h²

⇒ (5x)² = r² + (4x)²

⇒ 25x² = r² + 16x²

⇒ r² = 9x²

⇒ r = 3x

Volume (V) = 12936 cm³

⇒ \(\frac{1}{3}\)\(\pi\)r²h = 12936

⇒ \(\frac{1}{3}\) × \(\frac{22}{7}\) × (3x)² × 4x = 12936

⇒ \(\frac{1}{3}\) × \(\frac{22}{7}\) × 9x² × 4x = 12936

⇒ 264x³ / 7 = 12936

⇒ x³ = (12936 × 7) / 264

⇒ x³ = 343

⇒ x = 7

r = 3x = 3 × 7 = 21 cm

Diameter = 2r = 2 × 21 = 42 cm

∴ The correct answer is Option (1).