Final answer:
Using the principle of similar triangles, the height of a fence that casts a 14-foot shadow is also 14 feet, given that an 11-foot structure nearby casts an 11-foot shadow.
Step-by-step explanation:
The question asks us to determine the height of a fence given that it casts a 14-foot shadow while an 11-foot structure casts an 11-foot shadow. This is a problem that involves similar triangles, since both the structure and the fence create triangles with the ground and their respective shadows.
First, we recognize that the structure and its shadow form sides of a right-angle triangle, as does the fence with its shadow. We know that the height of the structure is equal to the length of its shadow, so the ratio of height to shadow length for the structure is 1:1. This means it forms a 45-degree right-angle triangle. We can then assume that, since the sunlight angle is constant, the fence will also form a similar triangle and have the same ratio of height to shadow length.
So we set up our proportion using the height and shadow length of the structure as:
height of structure / shadow of structure = height of fence / shadow of fence
Substituting the known values, we get:
11 feet / 11 feet = height of fence / 14 feet
By cross-multiplying, we find that the height of the fence is:
11 feet (height of fence) = 11 feet (14 feet)
Therefore, the height of the fence is also 14 feet, since the 11 cancels out on both sides of the proportion.