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 4cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (i) the perimeter (ii) the area of the rectangle.

Given the length of the rectangle is x cm and width of the rectangle is y cm.


As per given criteria, length is decreasing at the rate of 5cm/min



And width is increasing at the rate of 4cm/min



(i) Let P be the perimeter of the rectangle


And we know,


P = 2(x + y)


Differentiating both sides with respect to t, we get




Substituting the values from equation (i) and (ii),we get



When x = 8 cm and y = 6 cm, the rates of change of the perimeter is - 2cm/min (it is decreasing in nature) and is independent on length and width of the rectangle.


(ii) Let A be the area of the rectangle


And we know,


A = xy


Differentiating both sides with respect to t, we get



Now will apply the product rule of differentiation, i.e.,


, so the above equation becomes



Substituting the values from equation (i) and (ii),we get



When x = 8 cm and y = 6 cm, the above equation becomes,




When x = 8 cm and y = 6 cm, the rates of change of the area is 2cm/min (it is increasing in nature) and is dependent on length and width of the rectangle


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