Application of Derivatives

Question

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.

Solution

Let R be radius and l be the slanted height of cone.
Let OD = x, OA = OB = OC = 12, where 12 is a radius of sphere.
$In space rt. space space angle straight d space increment ODC comma space space space space straight R space equals space square root of 144 minus straight x squared end root In space rt space angle straight d space increment ADC comma space space space space space space space space l space equals space square root of straight R squared plus left parenthesis straight x plus 12 right parenthesis squared end root space equals space square root of 144 minus straight x squared plus straight x squared plus 144 plus 24 straight x end root space space space space space space space space space equals square root of 288 plus 24 straight x end root space equals space square root of 24 space equals space square root of straight x plus 12 end root$

Let S be curved surface area of cone.
$therefore space space straight S space equals space πR l space equals space straight pi square root of 144 minus straight x squared end root space square root of 24 space square root of straight x plus 12 end root space equals space straight pi square root of 24 space left parenthesis straight x plus 12 right parenthesis thin space square root of 12 minus straight x end root space$
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 $equals space straight pi square root of 24 open square brackets fraction numerator 12 minus 3 straight x over denominator 2 square root of 12 minus straight x end root end fraction close square brackets$
For S to be maximum or minimum, $dS over dx equals 0$
$therefore space space space space straight pi square root of 24 space open square brackets fraction numerator 12 minus 3 straight x over denominator 2 square root of 12 minus straight x end root end fraction close square brackets space equals 0 space space space space rightwards double arrow space space space straight x equals space 4 When space straight x less than 4 space left parenthesis slightly right parenthesis comma space space dS over dx space equals space plus ve$
$and space when space straight x thin space greater than 4 space left parenthesis slightly right parenthesis comma space dS over dx space equals space minus ve$
$therefore space space at space straight x space equals space 4 comma space space space dS over dx space changes space from space plus ve space to space minus ve therefore space space straight S space is space maximum space at space straight x space equals space 4 therefore space space space space altitude space AD space equals space 4 plus 12 space equals space 16.$

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Application Of Derivatives

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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