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
\(\therefore \frac{dx}{dt}\) = -5cm/min ...(i)
And width is increasing at the rate of 4cm/min
\(\therefore \frac{dx}{dt}\) = 4cm/min ...(ii)
(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
\( \frac{dA}{dt}=d(\frac{xy}{dt})\)
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
