Question

The length \(x\) of a rectangle is decreasing at the rate of 3 cm/min and the breadth \(y\) is increasing at the rate of 2 cm/min. When \(x=5\) cm and \(y=3\) cm, find the rate of change of the area of the rectangle.

Hint:

A = xy; use the product rule dA/dt = x·dy/dt + y·dx/dt with dx/dt negative.

Answer 1

Step 1: Modelling as functions of time:
Near the given instant, \(x(t)\) is decreasing and \(y(t)\) is increasing; write \(A(t)=x(t)y(t)\).

Step 2: Product rule with signed rates:
\(A'(t)=x'(t)y(t)+x(t)y'(t)\) with \(x'=-3\), \(y'=+2\) at this instant.

Step 3: Plugging in x=5, y=3:
\(A'=(-3)(3)+(5)(2)=-9+10=1\).

Final Answer:
The area is increasing at \(1\ \text{cm}^2/\text{min}\).\[ \boxed{1\ \text{cm}^2/\text{min}} \]