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Application Of Derivatives

Question
CBSEENMA12035509
Wired Faculty App

Two sides of a triangle are given. Find the angle between them such that the area shall be maximum.

Solution

Let straight theta be the angle between two given sides of lengths a and b
therefore space space space space space space increment space equals space 1 half straight a space straight b space sin space straight theta
where increment is area of triangle
             fraction numerator straight d increment over denominator dθ end fraction space equals space 1 half space straight a space straight b space cos space straight theta
Now,    fraction numerator straight d increment over denominator dθ end fraction equals space 0 space space space space space space space space space space space rightwards double arrow space space space 1 half space straight a space straight b space cos space straight theta space equals space 0 space space space space space rightwards double arrow space space space cos space straight theta space equals space 0
rightwards double arrow space space space space space straight theta space equals space straight pi over 2 space space space space space space space fraction numerator straight d squared increment over denominator dθ squared end fraction space equals space 1 half space straight a space straight b space sin space straight theta when space straight theta space equals space straight pi over 2. space fraction numerator straight d squared increment over denominator dθ squared end fraction space equals space minus 1 half space ab space sin space straight pi over 2 space equals space minus 1 half space ab space less than 0 therefore space space space space increment space space space is space maximum space when space straight theta space equals space straight pi over 2 space equals space 90 degree therefore space space space space required space angle space equals space 90 degree.

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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