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A cone has a height of 30 cm. A plane parallel to the base cuts off the top of a tiny cone. At what height above the base, the section has been formed if its volume $127$ the specified cone?

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20 cm

30 cm

40 cm

50 cm

answer is A.

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Detailed Solution

We have given a cone with a height of 30 cm such that a plane parallel to the base cuts off the top of a tiny cone.
We know that the volume of the cone is

13 π R 2 H,

where R is the radius of the base and H is the cone's height.
Consider the figure,
Question Image

Since the two triangles,

Δ A P C

and

Δ A Q E

are similar to each other, therefore we have,

r R = h H ⇒ r R = h 30 ∵ H = 30 c m ⇒ r = R h 30 ....... (1)

It is given that the volume of the small cone is

127

of the volume of the given cone.

∴ 13 π r 2 h = 127 × 13 R 2 × 30 ⇒ 13 π h R 30 2 h = 181 π R 2 × 30 ⇒ 13 π h 2 R 2 900 = 181 π R 2 × 30 ⇒ h 3 = 1 × 30 × 900 × 3 81

⇒ h 3 = 1000 cm 3 ⇒ h = 10 cm

From the figure,

P Q = A Q − A P ⇒ P Q = 30 − 10 cm ⇒ P Q = 20 cm

Therefore, the required height is 20 cm.
Hence, option (1) is correct.

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