Find the maximum lateral surface (curved surface) area in of a cylinder inscribed in a cone which has radius and height as
Give your answer to 2 decimal places.
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Imagine the figure to be 2-Dimensional with the cone to be a triangle and the cylinder to be a rectangle.
Since
△
A
D
C
,
△
A
M
G
and
△
G
H
C
are similar to each other,using similarity properties we get
r h = x h − y = r − x y
From the first and the last term we get y = r h ( r − x )
Since the lateral surface area of a cylinder is 2 π x y = r 2 π x h ( r − x ) .
Since the r 2 π h is constant,we bring it out. L . S . A = r 2 π h ( r x − x 2 )
So we have to only differentiate ( r x − x 2 ) and equate it to 0 so as to maximize the L . S . A .
After differentiating,we get r − 2 x = 0 ⇒ x = 2 r and plugging this in the equation we found earlier,we get y = 2 h .
Therefore,the lateral surface area of the cylinder = 2 π x y = 4 π = 1 2 . 5 7 .