Mathematics > Calculus > Differential Equations

Case Study 2

Camphor is a waxy, colourless solid with a strong aroma that evaporates through sublimation when left in open air at room temperature.

A cylindrical camphor tablet whose height equals its radius ($r$) evaporates such that the rate of reduction of its volume is proportional to its total surface area: $\dfrac{dV}{dt} = kS$, where $V$ is volume, $S$ is surface area, and $t$ is time in hours.

Based upon the above information, answer the following questions:

(i) Write the order and degree of the given differential equation. [1 mark]

(ii) Substituting $V = \pi r^3$ and $S = 2\pi r^2$, the differential equation $\dfrac{dr}{dt} = \dfrac{2}{3}k$ is obtained. Solve it, given that $r(0) = 5$ mm. [1 mark]

(iii)(a) If $r = 3$ mm when $t = 1$ hour, find the value of $k$. Hence find $t$ when $r = 0$ mm. [2 marks]

OR

(iii)(b) If $r = 1$ mm when $t = 1$ hour, find the value of $k$. Hence find $t$ when $r = 0$ mm. [2 marks]