Question

If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.

Hint:

To find the rate of change of the perimeter, differentiate the perimeter formula with respect to time and substitute the given rates of change of the dimensions.

Answer 1

The perimeter \( P \) of a rectangle is given by: \[ P = 2x + 2y. \] We are given: \[ \frac{dx}{dt} = -3 \, \text{cm/min} \text{(length is decreasing)}, \] \[ \frac{dy}{dt} = 2 \, \text{cm/min} \text{(width is increasing)}. \] To find the rate of change of the perimeter, differentiate the perimeter formula with respect to \( t \): \[ \frac{dP}{dt} = 2 \frac{dx}{dt} + 2 \frac{dy}{dt}. \] Substitute the given values: \[ \frac{dP}{dt} = 2(-3) + 2(2) = -6 + 4 = -2 \, \text{cm/min}. \] Conclusion: The rate of change in the perimeter of the rectangle is \( \boxed{-2} \, \text{cm/min} \), indicating that the perimeter is decreasing at the rate of 2 cm/min.