We are given that the length of the side of a square sheet of metal is increasing at the rate of 4 cm/sec. We need to find the rate at which the area of the sheet is increasing when the length of the sides is 2 cm.

Step-by-step solution:

1. Let the side of the square be $s$ (in cm).

   • The rate at which the side is increasing is: $\frac{ds}{dt} = 4 \, \text{cm/sec}$

2. The area $A$ of the square is given by the formula: $A = s^2$

3. Differentiate the area with respect to time $t$ to find the rate of change of area: $\frac{dA}{dt} = 2s \cdot \frac{ds}{dt}$ This equation relates the rate of change of the area $\frac{dA}{dt}$ to the rate of change of the side $\frac{ds}{dt}$.

4. Substitute the given values:

   • $s = 2 \, \text{cm}$
   • $\frac{ds}{dt} = 4 \, \text{cm/sec}$

   Now, calculate the rate of change of the area: $\frac{dA}{dt} = 2(2) \cdot 4 = 16 \, \text{cm}^2/\text{sec}$

Thus, the rate at which the area of the sheet is increasing when the side length is 2 cm is 16 cm²/sec.

Would you like a more detailed explanation of any part of this process?

Related questions:

1. What is the rate of change of the perimeter of the square in this case?
2. How would the rate of change of the area differ if the rate of change of the side length were 6 cm/sec?
3. If the side length was increasing at a constant rate, how would you find the total increase in area over a specific time?
4. How does the rate of change of the area depend on the size of the side of the square?
5. Can you generalize this process for a cube instead of a square, finding the rate of change of the volume?

Tip:

In problems involving rates of change, always differentiate the key relationships (like area or volume formulas) with respect to time and then substitute the known rates and values.