Final answer:
Option b) 40/48 = 9.6/x is not a valid proportion to find the height of the building because it does not compare corresponding sides of similar triangles. Valid proportions to solve for the building's height would set the building's shadow length in relation to the flagpole's shadow length and the known heights.
Step-by-step explanation:
The task is to find the height of the building using similar triangles formed by the building and its shadow and the flagpole and its shadow.
To find a valid proportion, the ratios should compare corresponding sides of the similar triangles.
A valid proportion compares the shadow of the building to its height or the shadow of the flagpole to its height.
Therefore, using the information that a 40-foot-tall flagpole casts a 9.6-foot-long shadow, while the building casts a 48-foot-long shadow, we can determine that choices a) 48/40 = 9.6/x, c) 9.6/48 = x/40, and d) 48/9.6 = x/40 are valid proportions to solve for the height of the building (x).
However, option b) 40/48 = 9.6/x is not valid because it compares the height of the flagpole to the length of the building's shadow, which are not corresponding sides.
When we correctly set up a valid proportion like 48/9.6 = x/40 and cross multiply, we get a linear equation that can be solved for x, the height of the building.