A mathematical model is posed for diffusion through a semi-infinite medium into an axisymmetric root that grows along its axis with a prescribed velocity. It is assumed that the concentration gradient at the root surface is proportional to the local concentration and linear in the rate of growth normal to the surface. An asymptotic solution, based on the hypotheses that the root is slender and that the growth rate is small compared with the radial-diffusion rate, culminates in an integral equation. An approximate solution to this integral equation is determined. The total uptake of the diffusing substance by the root is calculated, and a simple upper bound to this uptake is determined from the approximate solution to the integral equation. It appears that the approximate solution generally should be rather close to the exact solution for the model posed, but the validity of the assumed boundary condition at the root surface remains to be established.