To find the rate at which the area of the square is increasing, we need to use the relationship between the side length of the square, its area, and the rate of change of these quantities.
Let’s denote the side length of the square as s. The area A of the square can be expressed as:
A = s²
We know that the side length is increasing at a rate of 6 cm per second, which can be represented as:
ds/dt = 6 cm/s
To find the rate of change of the area with respect to time (dA/dt), we can differentiate the area formula with respect to time:
dA/dt = 2s * (ds/dt)
Next, we need to find the value of s when the area is 16 cm²:
A = s² = 16
Taking the square root of both sides, we get:
s = √16 = 4 cm
Now, we can substitute the values of s and ds/dt into the differentiated area formula:
dA/dt = 2 * 4 cm * 6 cm/s
This simplifies to:
dA/dt = 48 cm²/s
Therefore, the area of the square is increasing at a rate of 48 cm² per second when the area is 16 cm².