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Application Of Derivatives

Question

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Find the area of greatest rectangle which can be inscribed in a circle of radius 10 cm.

Solution

Let O be the centre of circle of radius 10 cm. Let ABCD be the rectangle inscribed in the circle such that AB = x, AD = y

Now comma space space space space AB squared plus BC squared space equals AC squared space space space space space space rightwards double arrow space space space straight x squared plus straight y squared space equals space 400

...(1)

Let space straight P space be space the space area space of space rectangle

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dP over dx space equals straight x. space fraction numerator negative 2 straight x over denominator 2 square root of 400 minus straight x squared end root end fraction plus square root of 400 minus straight x squared end root space. space 1 space space space space space space space space space space space space equals negative fraction numerator straight x squared over denominator square root of 400 minus straight x squared end root end fraction plus fraction numerator square root of 400 minus straight x squared end root over denominator 1 end fraction space space space space space space space space space space space space equals fraction numerator negative straight x squared plus 400 minus straight x squared over denominator square root of 400 minus straight x squared end root end fraction space equals space fraction numerator 400 minus 2 straight x squared over denominator square root of 400 minus straight x squared end root end fraction space space dP over dx space equals space 0 space space space space rightwards double arrow space space space space fraction numerator 400 minus 2 straight x squared over denominator square root of 400 minus straight x squared end root end fraction space equals space 0 space space space space space space rightwards double arrow space space space space 400 minus 2 straight x squared space equals space 0 space space space space rightwards double arrow space space space space straight x squared space equals space 200 rightwards double arrow space space space space straight x space equals space 10 square root of 2 comma space space space as space straight x space is space positive fraction numerator straight d squared straight P over denominator dx squared end fraction space equals space fraction numerator square root of 400 minus straight x squared end root. space space left parenthesis negative 4 straight x right parenthesis space minus space left parenthesis 400 minus 2 straight x squared right parenthesis. space begin display style fraction numerator negative 2 space straight x over denominator 2 space square root of 400 minus straight x squared end root end fraction end style over denominator 400 minus straight x squared end fraction

When space straight x space equals space 10 square root of 2 comma space space fraction numerator straight d squared straight P over denominator dx squared end fraction space equals space fraction numerator square root of 400 minus 200 end root. space left parenthesis negative 40 square root of 2 right parenthesis space minus space left parenthesis 400 minus 400 right parenthesis. space begin display style fraction numerator 2 cross times 10 square root of 2 over denominator 2 square root of 400 minus 200 end root end fraction end style over denominator 400 minus 200 end fraction space space space space space space space space space space space space space space equals negative fraction numerator 800 minus 0 over denominator 200 end fraction space equals space minus 4 space less than space 0 therefore space space space space space space straight P space is space maximum space when space straight x space equals space 10 square root of 2 comma space space space space straight y space equals space square root of 400 minus 200 end root space equals space square root of 200 space equals space 10 square root of 2 therefore space space space space space space area space of space rectangle space space equals space left parenthesis 10 square root of 2 right parenthesis space left parenthesis 10 square root of 2 right parenthesis space equals space 200 space sq. space 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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