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State true or false.

The formula for the volume of the frustum of a cone is $13 π h (r 12 + r 22 + r 1 r 2)$ , given to you in the figure below is, using the symbols as explained.

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True

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answer is A.

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Detailed Solution

Given is a frustum of a cone with h as height, l as the slant height,

, r 1

and

r 2

radii of the ends where

r 1,

r 2

Derive the formula for the volume of the frustum of a cone, given to you in Section 13.5,using the symbols as explained

Extend the side BC and AD of the frustum of the cone to meet at O.
From the figure:
The frustum of a cone can be viewed as a difference of two right circular cones OAB and OCD.
Let

h 1

and

l 1

be the height and slant height of cone OAB and

h 2

and

l 2

be the height and slant height of cone OCD respectively.

Δ A P O

a n d Δ D Q O

As both cones are right angled,

Δ A P O = Δ D Q O = 900 ∠ A O P = ∠ D O Q (C o m m o n a n g l e) T h e r e f o r e, Δ A P O ~ Δ D Q O A . A c r i t e r i o n A P D Q = A O D O = O P O Q (b y C P C T)

Put the values:

r 1 r 2 = l 1 l 2 = h 1 h 2 ⇒ r 1 r 2 = h 1 h 2 ⇒ r 2 r 1 = h 2 h 1

Subtract 1 from both sides again :
Question Image

Volume of frustum of cone = Volume of cone OAB - Volume of cone OCD:

⇒ V o l u m e = 13 π r 12 h 1 − 13 π r 22 h 2 ⇒ V o l u m e = 13 π r 12 h 1 − r 22 h 2 ⇒ V o l u m e = 13 π r 12 × h r 1 r 1 − r 2 − r 22 × h r 2 r 1 − r 2

Simplify the equation:
Question Image

Hence, volume of the frustum of a cone

= 13 π h (r 12 + r 22 + r 1 r 2)

Therefore, the given statement is true.
Therefore, the correct answer is option (1).

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