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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.

Let ABCD be given trapezium such that BC = CD = DA = 10 cm

From D, draw DP X ⊥ and from C, draw CQ ⊥ AB
so that PQ = 10 cm.
Now ∆APD ≡ ∆QBC
AP = QB = x cm. say.
In rt. ∠ d ∆APD.
DP² = AD² – AP³ = 100 – x² ⇒ DP = $square root of 100 minus straight x squared end root$
Also $CQ space equals space DP space equals space square root of 100 minus straight x squared end root$
Let y be area of trapezium.
$therefore space space space space space space space space space space space straight y space equals space 1 half left parenthesis sum space of space parallel space sides right parenthesis space left parenthesis height right parenthesis space space space space space space space space space space space space space space space space space space space equals space 1 half left parenthesis 2 straight x plus 10 plus 10 right parenthesis space square root of 100 minus straight x squared end root therefore space space space space space space space space space space space space space space space space space space space straight y space equals space left parenthesis straight x plus 10 right parenthesis space square root of 100 minus straight x squared end root therefore space space space space space space space space space space space space space dy over dx space equals space left parenthesis straight x plus 10 right parenthesis space straight d over dx left parenthesis square root of 100 minus straight x squared end root right parenthesis space plus space left parenthesis square root of 100 minus straight x squared end root right parenthesis. space straight d over dx left parenthesis straight x plus 10 right parenthesis$
   $equals space left parenthesis straight x plus 10 right parenthesis. space space space fraction numerator negative 2 straight x over denominator 2 square root of 100 minus straight x squared end root end fraction plus square root of 100 minus straight x squared end root. space 1$
   $equals space minus fraction numerator straight x left parenthesis straight x plus 10 right parenthesis over denominator square root of 100 minus straight x squared end root end fraction plus fraction numerator square root of 100 minus straight x squared end root over denominator 1 end fraction space equals space fraction numerator negative straight x left parenthesis straight x plus 1 right parenthesis space plus 100 space minus straight x squared over denominator square root of 100 minus straight x squared end root end fraction space equals space fraction numerator negative straight x squared minus 10 straight x plus 100 minus straight x squared over denominator square root of 100 minus straight x squared end root end fraction$
$therefore space space space space space space dy over dx space equals space fraction numerator negative 2 straight x squared minus 10 straight x plus 100 over denominator square root of 100 minus straight x squared end root end fraction Now comma space space dy over dx space equals space 0 space space space space space space space space rightwards double arrow space space space fraction numerator negative 2 straight x squared minus 10 straight x plus 100 over denominator square root of 100 minus straight x squared end root end fraction space equals space 0 rightwards double arrow space space minus 2 straight x squared minus 10 straight x plus 100 space equals 0 space space space space space space space space space rightwards double arrow space space space straight x squared plus 5 straight x minus 50 space equals space 0 therefore space space space space space space space left parenthesis straight x minus 5 right parenthesis left parenthesis straight x plus 10 right parenthesis space equals space 0 space space space space space space space space rightwards double arrow space space space space straight x space equals space 5 comma space space minus 10$
Rejecting x = -10 as distance x cannot be negative, we get, x = 5
$fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator square root of 100 minus straight x squared end root. space begin display style straight d over dx end style left parenthesis negative 2 straight x squared minus 10 straight x plus 100 right parenthesis space minus left parenthesis negative 2 straight x squared minus 10 straight x plus 100 right parenthesis begin display style straight d over dx end style left parenthesis square root of 100 minus straight x squared end root right parenthesis over denominator open parentheses square root of 100 minus straight x squared end root close parentheses squared end fraction space space space space space space space space space space space space equals fraction numerator square root of 100 minus straight x squared. end root space space left parenthesis negative 4 straight x minus 10 right parenthesis space minus space left parenthesis negative 2 straight x squared minus 10 straight x plus 100 right parenthesis space begin display style fraction numerator negative 2 straight x over denominator 2 square root of 100 minus straight x squared end root end fraction end style over denominator 100 minus straight x squared end fraction space space space space space space space space space space space equals space fraction numerator begin display style fraction numerator left parenthesis negative 4 straight x minus 10 right parenthesis thin space left parenthesis square root of 100 minus straight x squared end root right parenthesis over denominator 1 end fraction end style plus begin display style fraction numerator straight x left parenthesis negative 2 straight x squared minus 10 straight x plus 100 right parenthesis over denominator square root of 100 minus straight x squared end root end fraction end style over denominator 100 minus straight x squared end fraction space space space$

$equals space fraction numerator left parenthesis negative 4 straight x minus 10 right parenthesis thin space left parenthesis 100 minus straight x squared right parenthesis space plus straight x left parenthesis negative 2 straight x squared minus 10 straight x plus 100 right parenthesis over denominator left parenthesis 100 minus straight x squared right parenthesis begin display style 3 over 2 end style end fraction equals space fraction numerator negative 400 straight x plus 4 straight x cubed minus 1000 plus 10 straight x squared minus 2 straight x cubed minus 10 straight x squared plus 100 straight x over denominator left parenthesis 100 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction therefore space space space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator 2 straight x cubed minus 300 straight x minus 100 over denominator left parenthesis 100 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction When space straight x space equals space 5 comma space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator 2 left parenthesis 5 right parenthesis cubed minus 300 left parenthesis 5 right parenthesis space minus 1000 over denominator left parenthesis 100 minus 25 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space equals space fraction numerator negative 2250 over denominator 75 square root of 75 end fraction less than 0 therefore space space space space straight y space is space maximum space when space straight x space equals space 5 and space maximum space area space space equals space left parenthesis 5 plus 10 right parenthesis space square root of 100 minus 25 end root space equals space 15 square root of 75 space equals space 15 space cross times 5 square root of 3 space equals space 75 square root of 3 space cm squared space space space space space space space space$