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The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x is equal to 8 cm and yis equal to 6 cm, find the rates of change of the perimeter,

7. The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find  the rate of change of (a) the perimeter of rectangle
 

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Given =   Length x of a rectangle is decreasing at the rate (\frac{dx}{dt})  =   -5  cm/minute                  (-ve sign indicates decrease in rate)
                the width y is increasing at the rate   (\frac{dy}{dt}) = 4 cm/minute
To find =  \frac{dP}{dt}    and      at  x = 8 cm and y = 6 cm                     , where P is perimeter 
Solution:-

Perimeter of rectangle(P) = 2(x+y)
\frac{dP}{dt} = \frac{d(2(x+y))}{dt} = 2\left ( \frac{dx}{dt} + \frac{dy}{dt} \right ) = 2(-5+4) = -2 \ cm/minute
Hence, Perimeter decreases at the rate of  2 \ cm/minute 
 

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