You need to construct an open$-$top rectangular box with a square base that must hold a volume of exactly $725c{m}^3.$ The material for the base of the box costs $7$ cents $\frac{?}{c{m}^2}$ and the material for the sides of the box costs $5$ cents $\frac{?}{c{m}^2}.$

Find the dimensions for a box that will minimize the cost of the materials used to construct box. Please show your answers to at least $4$ decimal places.

width: $$ cm

height: $$ cm