1. **State the problem:** We have a rectangular sheet of metal with width $w=200$ mm and length $l=300$ mm. The width is increasing at a rate of $\frac{dw}{dt} = 2$ mm/min and the length at $\frac{dl}{dt} = 3$ mm/min. We want to find the rate of change of the area $A$ at that instant. 2. **Formula:** The area of a rectangle is given by $$A = w \times l$$ To find the rate of change of area with respect to time, differentiate both sides with respect to $t$: $$\frac{dA}{dt} = w \frac{dl}{dt} + l \frac{dw}{dt}$$ This uses the product rule for differentiation. 3. **Substitute known values:** $$\frac{dA}{dt} = 200 \times 3 + 300 \times 2$$ 4. **Calculate:** $$\frac{dA}{dt} = 600 + 600 = 1200$$ 5. **Interpretation:** The area is increasing at a rate of 1200 square millimeters per minute at that instant. **Final answer:** 1200