Two nearby trees are perpendicular to the ground, which is flat. One of these trees is 10 feet tall and has a shadow that is 5 feet long. At the same time, the shadow of the other tree is 2 feet long. How tall, in feet, is the other tree?
The correct answer and explanation is:
The height of the other tree is 4 feet.
Explanation:
The problem involves similar triangles formed by the trees and their shadows. When the ground is flat and the sunlight hits the trees at the same angle, the ratios of the heights of the trees to the lengths of their shadows are equal.
This proportional relationship can be expressed as follows: Height of Tree 1Shadow of Tree 1=Height of Tree 2Shadow of Tree 2\frac{\text{Height of Tree 1}}{\text{Shadow of Tree 1}} = \frac{\text{Height of Tree 2}}{\text{Shadow of Tree 2}}
Step-by-Step Solution:
- Identify the known quantities:
- Height of Tree 1: 10 feet10 \, \text{feet}
- Shadow of Tree 1: 5 feet5 \, \text{feet}
- Shadow of Tree 2: 2 feet2 \, \text{feet}
- Let hh represent the unknown height of Tree 2. Using the proportional relationship:
105=h2\frac{10}{5} = \frac{h}{2}
- Simplify the ratio of Tree 1:
105=2\frac{10}{5} = 2
- Substitute 22 for 105\frac{10}{5} in the equation:
2=h22 = \frac{h}{2}
- Solve for hh:
h=2×2h = 2 \times 2 h=4h = 4
Final Answer:
The height of the other tree is 4 feet.
Why This Works:
The principle of similar triangles ensures that the ratios of corresponding sides are equal when the triangles share the same angles. In this problem, both trees are perpendicular to the ground, and the sunlight creates identical angles for both trees and their shadows. This allows us to directly compare their height-to-shadow ratios, making the calculation straightforward and reliable.
This approach not only provides the solution but also illustrates the consistency of mathematical reasoning in real-world scenarios, such as understanding proportions and scaling relationships.