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Application Of Derivatives

Question

Wired Faculty App

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5 square root of 3 space cm space is space 500 space straight pi space cm cubed.$

Solution

Let h be the height and R be the base radius of the inscribed cylinder.
r, radius of sphere =

5 square root of 3 space cm

In

increment OAC comma

OC squared plus CA squared space equals space OA squared space space space space space rightwards double arrow space space space space space space open parentheses straight h over 2 close parentheses squared plus space straight R squared space equals space left parenthesis 5 square root of 3 right parenthesis squared therefore space space space space straight R squared space equals space 75 space minus space straight h squared over 4 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

Let V be volume of the cylinder.

therefore space space space space space space space straight V space equals space πR squared straight h space equals space straight pi space open parentheses 75 minus straight h squared over 4 close parentheses straight h space equals space straight pi over 4 left parenthesis 300 minus straight h squared right parenthesis space straight h space equals space straight pi over 4 left parenthesis 300 minus straight h cubed right parenthesis space space space space space space dV over dh space equals space straight pi over 4 left parenthesis 300 minus 3 straight h squared right parenthesis. space space space space space space space dV over dh space equals space 0 space space space space space space space space space space rightwards double arrow space space space space space straight pi over 4 left parenthesis 300 minus 3 straight h squared right parenthesis space equals space 0 space space space space space space rightwards double arrow space space 300 minus 3 straight h squared space equals space 0 space space rightwards double arrow space space space space space space space straight h squared space equals space 100 space space space space space space space space space space space space rightwards double arrow space space space space space straight h space equals space 10 space space space space space space space space space space space space space space space space space space space space fraction numerator straight d squared straight V over denominator dh squared end fraction space equals space straight pi over 4 left parenthesis negative 6 space straight h right parenthesis space equals negative fraction numerator 3 πh over denominator 2 end fraction

When space space space space space space space straight h space equals space 10 comma space space space fraction numerator straight d squared straight V over denominator dh squared end fraction space equals space minus fraction numerator 3 straight pi over denominator 2 end fraction cross times 10 space equals space minus 15 space straight pi space less than 0 therefore space space space space straight V space is space maximum space when space straight h space equals space 10 therefore space space space space space greatest space volume space equals space straight pi over 4 left parenthesis 300 minus 100 right parenthesis thin space left parenthesis 10 right parenthesis space equals space straight pi over 4 cross times 200 space cross times space 10 space equals space 500 space straight pi space cm cubed.

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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