It is given that the radius $R$, in meters, of the expansion of a liquid in the soil is given by $t$ (time elapsed since the liquid was released), the mass $M$ of the liquid released and of the variable $D$, conductivity of soil in $\text{kg}^{-1}\text m^5\text s^{-1}$.
The first part of the question is to find an expression of $R$ in terms of $M$, $t$ and $D$. By using dimensional analysis I find the expression $R=k(M\cdot t\cdot D)^{1/5}$, where $k$ is dimensionless.
If the particular liquid reaches a radius of $100\text m$ in $10$ days by using a mass $M_1$ of the liquid, in how much time will it reach a radius of $1000\text m$ given that we use $5$ times the mass of liquid that we used before ($M_2=5M_1$)?
By standard procedure I find the result to be approximately $548$ years. Is that reasonable and/or correct?
Indeed, by following the general guidelines the answer is 548 years (approx) which is correct.