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: Setting up the area formula:
Area \(A=xy\).

Step 2: Differentiating w.r.t. time:
By the product rule, \(\dfrac{dA}{dt}=x\dfrac{dy}{dt}+y\dfrac{dx}{dt}\).

Step 3: Substituting given values:
Here \(\dfrac{dx}{dt}=-3\) cm/min (decreasing), \(\dfrac{dy}{dt}=2\) cm/min (increasing), \(x=5\), \(y=3\): \(\dfrac{dA}{dt}=5(2)+3(-3)=10-9\).

Final Answer:
\[ \boxed{\dfrac{dA}{dt}=1\ \text{cm}^2/\text{min (increasing)}} \]