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Application Of Derivatives

Question

Wired Faculty App

Find the largest possible area of a right-angled triangle whose hypotenuse is 5 cm long.

Solution

Let x cm and y cm be two sides of a right angled triangle with hypotenuse 5 cm.

therefore space space space straight x squared plus straight y squared space equals space 25 space space space rightwards double arrow space space space straight y squared space equals space 25 minus straight x squared therefore space space space space straight y space equals space square root of 25 minus straight x squared end root 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 space space space space space space space space space space space space.... left parenthesis 1 right parenthesis

Let A be area of triangle

therefore space space space space straight A space equals space 1 half xy space equals space 1 half straight x square root of 25 minus straight x squared end root space space space space space space open square brackets because space space of space left parenthesis 1 right parenthesis close square brackets therefore space space space space space dA over dx space equals 1 half open square brackets straight x. space fraction numerator negative 2 straight x over denominator 2 square root of 25 minus straight x squared end root end fraction plus square root of 25 minus straight x squared end root.1 close square brackets space space space space space space space space space space space space space space space space space space space equals 1 half open square brackets negative fraction numerator straight x squared over denominator square root of 25 minus straight x squared end root end fraction plus fraction numerator square root of 25 minus straight x squared end root over denominator 1 end fraction close square brackets

equals space 1 half open square brackets fraction numerator negative straight x squared plus 25 minus straight x squared over denominator square root of 25 minus straight x squared end root end fraction close square brackets space equals space 1 half open parentheses fraction numerator 25 minus 2 straight x squared over denominator square root of 25 minus straight x squared end root end fraction close parentheses

For maximum or minimum values,

dA over dx space equals space 0

therefore space space space space 1 half open parentheses fraction numerator 25 minus 2 straight x squared over denominator square root of 25 minus straight x squared end root end fraction close parentheses space equals space 0 space space space space space space space space space space space space rightwards double arrow space space space 25 minus 2 straight x squared space equals space 0 rightwards double arrow space space space space space space straight x squared space equals space 25 over 2 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 rightwards double arrow space space space space straight x space equals space fraction numerator 5 over denominator square root of 2 end fraction as space straight x space cannot space be space negative fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space 1 half open square brackets fraction numerator square root of 25 minus straight x squared end root. space left parenthesis negative 4 straight x right parenthesis space minus space left parenthesis 25 minus 2 straight x right parenthesis squared. space begin display style fraction numerator negative 2 straight x over denominator 2 square root of 25 minus straight x squared end root end fraction end style over denominator 25 minus straight x squared end fraction close square brackets space space space space space space space space space space space space equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis end fraction open square brackets negative 4 straight x square root of 25 minus straight x squared end root plus fraction numerator straight x left parenthesis 25 minus 2 straight x squared right parenthesis over denominator square root of 25 minus straight x squared end root end fraction close square brackets space space space space space space space space space space space space equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis end fraction open square brackets fraction numerator negative 4 straight x left parenthesis 25 minus straight x squared right parenthesis space plus space straight x left parenthesis 25 minus 2 straight x squared right parenthesis over denominator square root of 25 minus straight x squared end root end fraction close square brackets space space space space space space space space space space space space

equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis end fraction open square brackets fraction numerator negative 100 straight x plus 4 straight x cubed plus 25 straight x minus 2 straight x cubed over denominator square root of 25 minus straight x squared end root end fraction close square brackets equals space fraction numerator 1 over denominator 2 left parenthesis 25 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space left parenthesis 2 straight x cubed minus 75 straight x right parenthesis

When space straight x space equals space fraction numerator 5 over denominator square root of 2 end fraction. fraction numerator straight d squared straight A over denominator dx squared end fraction space equals space fraction numerator 1 over denominator 2 open parentheses 25 minus begin display style 25 over 2 end style close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction open square brackets fraction numerator 2 cross times 125 over denominator 2 square root of 2 end fraction minus 75 cross times fraction numerator 5 over denominator square root of 2 end fraction close square brackets space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 open parentheses begin display style 25 over 2 end style close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction open square brackets fraction numerator 125 over denominator square root of 2 end fraction minus fraction numerator 375 over denominator square root of 2 end fraction close square brackets space equals space fraction numerator 1 over denominator 2 open parentheses begin display style 25 over 2 end style close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction open square brackets negative fraction numerator 250 over denominator square root of 2 end fraction close square brackets less than 0

therefore space space space straight A space is space maximum space when space straight x space equals space fraction numerator 5 over denominator square root of 2 end fraction therefore space space space maximum space area space space equals space 1 half fraction numerator 5 over denominator square root of 2 end fraction square root of 25 minus 25 over 2 end root 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 equals fraction numerator 5 over denominator 2 square root of 2 end fraction cross times square root of 25 over 2 end root space equals space fraction numerator 5 over denominator 2 square root of 2 end fraction cross times fraction numerator 5 over denominator square root of 2 end fraction space equals space 25 over 4 space sq. space units. space

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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