A bottle has the following cross-section. The radius of the bigger cylinder is 6 cm, and that of the smaller one is 3 cm. The height of the frustum of the cone is 7 cm, which is also equal to the height of each of the two cylinders. The volume of the bottle is? [π = 22/7]
Explanation:
Total volume of the bottle = Volume of top cylinder + Volume of frustum + Volume of bottom cylinder
Now,
Volume of top cylinder = π × r2 × h = 22/7 × 32 × 7 = 198 cm2.
Volume of bottom cylinder = π × r2 × h = 22/7 × 62 × 7 = 792 cm2.
Now, for the frustum of the cone, BCED,
∆ADE ~ ∆ABC ∴ AX : AY = DE : BC = 3 : 6 = 1 : 2 ⇒ AX = 7 and AY = 14
Volume of frustum = Volume of bigger cone (ABC) - Volume of smaller cone (ADE) = 1/3 × π × R2 × h - 1/3 × π × r2 × h = 1/3 × 22/7 × 62 × 14 - 1/3 × 22/7 × 32 × 7 = 528 - 66 = 462
∴ Total volume of the bottle = 198 + 792 + 462 = 1452 cm2.
Hence, option (b).
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