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Application of Derivatives

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Long Answer Type

351.

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

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352. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the maximum length of the hypotenuse is

open parentheses straight a to the power of 2 over 3 end exponent plus straight b to the power of 2 over 3 end exponent close parentheses to the power of 3 over 2 end exponent.

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353. Find the maximum-area of an isosceles triangle inscribed in the ellipse

straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1

with its vertex at one end of the major axis.

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Short Answer Type

354.

Let AP and BQ be two vertical poles at points A and B, respectively. If AP = 16 m. BQ = 22 m and AB = 20 m, then find the distance of a point R on AB from the point A such that RP² + RQ² is minimum.

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Long Answer Type

355.

If length of three sides of a trapezium other than base are equal to 10 cm. then find the area of the trapezium when it is maximum.

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Short Answer Type

356.
Use differentials to approximate $square root of 25.3 end root.$

Take $straight y equals space square root of straight x comma space space space straight x space equals space 25 space comma space space space dx space equals space δx equals space 0.3 space space so space that space straight x plus δx space equals space 25.3$
Then $δy space equals space square root of 25.3 end root space minus space square root of 25$ $open square brackets because space space space δy space equals space square root of straight x plus δx end root space minus square root of straight x close square brackets space$
$therefore space space space space square root of 25.3 end root space space equals space 5 space plus space δy$ ...(1)
Now, $δy$ is approximately equal to dy and
$dy space equals dy over dx dx space equals space fraction numerator 1 over denominator 2 square root of straight x end fraction left parenthesis 0.3 right parenthesis space equals space fraction numerator 1 over denominator 2 square root of 25 end fraction left parenthesis 0.3 right parenthesis space equals space fraction numerator 0.3 over denominator 10 end fraction space equals space 0.03$
$therefore space space space from space left parenthesis 1 right parenthesis comma space space space square root of 25.3 end root space equals space 5 plus 0.03 space equals space 5.03$