1. **State the problem:** We have a circular metal disk with radius $r$ expanding over time. When $r=40$ cm, the radius is increasing at a rate of $\frac{dr}{dt} = 0.02$ cm/s. We need to find the rate of increase of the area $A$ at this moment. 2. **Formula and rules:** The area of a circle is given by: $$ A = \pi r^2 $$ To find the rate of change of area with respect to time, we differentiate both sides with respect to $t$ using the chain rule: $$ \frac{dA}{dt} = 2\pi r \frac{dr}{dt} $$ 3. **Substitute known values:** Given $r=40$ cm and $\frac{dr}{dt} = 0.02$ cm/s, substitute into the formula: $$ \frac{dA}{dt} = 2 \pi \times 40 \times 0.02 $$ 4. **Calculate:** $$ \frac{dA}{dt} = 2 \pi \times 40 \times 0.02 = 2 \pi \times \cancel{40} \times \cancel{0.02} $$ Simplify the multiplication: $$ 40 \times 0.02 = 0.8 $$ So, $$ \frac{dA}{dt} = 2 \pi \times 0.8 = 1.6 \pi $$ 5. **Final answer:** The rate of increase of the area when the radius is 40 cm is: $$ \boxed{\frac{dA}{dt} = 1.6 \pi \text{ cm}^2/\text{s}} $$ This means the area is increasing at approximately $1.6 \times 3.1416 = 5.0265$ cm$^2$/s at that moment.