Root hair size determines the surface area/volume ratio of the whole roots exposed to the nutrient and water pools. thereby likely impacting nutrient and water uptake rates. The speed at which they grow is determined both by cell-intrinsic factors like hormones (e. g. https://www.ngetikin.com/quick-offer-1-Gallon-Universal-Primer-Pro-Deep-Penetrating-Primer-big-deal/