Full Answer Section

Solution:

Let the length of each edge of the ice cube be denoted by $l$. The volume of the cube, $V$, is given by

V = l^3

Since the edges are shrinking at a rate of 2 centimeters per minute, we write

\frac{dl}{dt} = -2

where $t$ represents time. The negative sign is there because $l$ is decreasing with time. To find the rate at which the volume change, dtdV​, we can use the chain rule.

\frac{dV}{dt} = 3l^2 \cdot \frac{dl}{dt} = 3l^2 (-2) = -6l^2

At the moment when the length of each edge is 15 cm, we have $l=15$. Therefore, the rate at which the volume is changing is

\frac{dV}{dt} = -6(15)^2 = -1350 cm^3/min

In other words, the volume of the ice cube is decreasing at a rate of 1350 cubic centimeters per minute.

Interpretation:

The negative sign in the expression for dtdV​ tells us that the volume is decreasing. The rate of change, $-1350$ cubic centimeters per minute, is quite large. This means that the ice cube is melting very quickly.

Remarks:

In the real world, the melting process would not be as simple as we have described it here. The edges of the ice cube would not shrink at a perfectly uniform rate, and the shape of the cube would change as it melted. However, the mathematical analysis presented here gives us a good approximation of the rate at which the volume of the ice cube is changing.