Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area is 16 cm²?

To find the rate at which the area of the square is increasing, we start with the formula for the area of a square, which is A = s², where s is the length of a side of the square.

Given that the area of the square is currently 16 cm², we can find the length of the side:

• A = 16 cm²
• s = √16 = 4 cm

We know the sides of the square are increasing at a rate of ds/dt = 6 cm/s. To find the rate at which the area is increasing, we use the concept of related rates.

We differentiate the area equation with respect to time:

dA/dt = 2s * ds/dt

Now, we can substitute in the values we know:

• s = 4 cm
• ds/dt = 6 cm/s

Plugging these values into the differentiated equation:

dA/dt = 2(4 cm)(6 cm/s)

dA/dt = 48 cm²/s

Thus, the area of the square is increasing at a rate of 48 cm²/s when the area is 16 cm².