Many materials, such as brick, steel, and concrete expand with increases in temperature. This is why spaces are placed between the cement slabs in sidewalks. Suppose you have a 100-ft length of material securely fastened at both ends and assume that the buckle is linear.(it is not, but this assumption will serve as a worthwhile approximation) if the height of the buckle is x ft and the percentage of swelling is y then x and y are related as shown in figure