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The height of a cone is 30 cm. A small cone is cut off at the top by a plane parallel to the base. If its volume be 1/27 the of the volume of the given cone, at what height above the base is the sectionm ade ?
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- 19 cm
- 20 cm
- 12 cm
- 15 cm
- 19 cm
Correct Option: B
Let H and R be the height and radius of bigger cone respectively and h and r that of smaller cone.
From triangles AOB and AMN. ∠A is common and MN || OB.
∴ Triangles AOB and AMN are similar,
| ∴ | = | ||
| AM | MN |
| ⇒ | = | ..........(i) | ||
| h | r |
| Volume of smaller cone = | πr²h | |
| 3 |
| Volume of bigger cone = | πR²h | |
| 3 |
∴ According to the question,
| πr²h = |  | πR²h |  | × | ||||
| 3 | 3 | 27 |
| ⇒ r²h = | ||
| 27 |
→ 27r²h = R²H
| ⇒ | = | ||
| H | r² |
| ⇒ | = | | | ² | ....[From(i)] | ||
| H | h |
| ⇒ | = | ||
| H | r² |
⇒ 27h³ = 900H = 900 × 30
| ⇒ h³ = | = 1000 | |
| 27 |
⇒ h = ³√1000 = 10 cm
∴ Required height = 30 – 10 = 20 cm
