Question

Suppose a (magical?) ice cube melts, and as it melts it remains in the shape of a perfect cube, of side length x, and the rate of change in its volume is proportional to its surface area, meaning dvolumedt=k⋅(surface area). Initially at time t=0, we have that x is 3 centimeters, but when t=25 minutes, so the cube has been melting for 25 minutes, then x is 2 centimeters. At what time t will I find that x is 1 centimeter?

Answer 1

This question is about the chain rule: you need to create an equation for x in terms of t so that you can substitute in the data you're given. First, given that the rate of volume decrease is proportional to the surface area, you get (as you said):
\[\frac{ dV }{ dt }=-kA\]
Then you'll need to express the volume (V) and surface area (A) of the cube in terms of x.
\[V = x^3, A = 6x^2\]
What you want to find is the rate of change of x, which means finding dx/dt. To do this, you need to use the chain rule:
\[\frac{ dx }{ dt}=\frac{ dx }{ dV }\times \frac{ dV }{ dt }\]
See if you can find the derivatives from what you have and then go from there. If you're stuck, just ask. :)

Answer 2

Thanks that helped very much!