Let's start by identifying the relevant formula for the surface area of a cube. The surface area (A) of a cube with side length (s) is:
A = 6s^2
We are given that the side length of the cube is changing at a rate of -2 cm/min (the negative sign indicates that the side length is decreasing). We want to find the rate of change of the surface area when the side length is 4 cm.
To solve this problem, we can use the chain rule of differentiation. We have:
dA/dt = dA/ds * ds/dt
where dA/dt is the rate of change of the surface area, dA/ds is the rate of change of the surface area with respect to the side length (which we can find by differentiating the surface area formula), ds/dt is the rate of change of the side length, and t is time.
Differentiating the surface area formula with respect to the side length, we get:
dA/ds = 12s
Plugging in s = 4 cm (since we want to find the rate of change when the side length is 4 cm), we get:
dA/ds = 12(4) = 48 cm^2
We are given that ds/dt = -2 cm/min (since the side length is decreasing at a rate of 2 cm per minute). Plugging in these values, we get:
dA/dt = (48 cm^2/cm) * (-2 cm/min) = -96 cm^2/min
Therefore, the rate of change of the surface area when the side of the durian is 4 cm is -96 cm^2/min. Note that the negative sign indicates that the surface area is decreasing at a rate of 96 cm^2 per minute.