Length of a Shadow If a tree 20 ft tall casts a shadow 8 ft long, how long would the shadow of a 30-ft tree be at the same time and place?

Verified step by step guidance

1. Step 1: Understand the problem involves similar triangles. The height of the tree and the length of its shadow form a right triangle.
2. Step 2: Set up a proportion using the known values. Let the length of the shadow of the 30-ft tree be x.
3. Step 3: Write the proportion based on the similar triangles: \( \frac{20}{8} = \frac{30}{x} \).
4. Step 4: Cross-multiply to solve for x: \( 20x = 8 \times 30 \).
5. Step 5: Solve for x by dividing both sides by 20: \( x = \frac{8 \times 30}{20} \).

Key Concepts

Similar Triangles

The concept of similar triangles states that if two triangles have the same shape, their corresponding angles are equal, and the lengths of their corresponding sides are proportional. In this problem, the heights of the trees and the lengths of their shadows form two similar triangles, allowing us to set up a proportion to find the unknown shadow length.

Proportional Relationships

Proportional relationships occur when two quantities maintain a constant ratio. In this scenario, the ratio of the height of the tree to the length of its shadow remains constant. By establishing this ratio from the first tree, we can apply it to the second tree to determine the length of its shadow.

Ratios and Fractions

Ratios express the relationship between two quantities, often written as a fraction. In this question, we can express the height of the trees and the lengths of their shadows as ratios. Understanding how to manipulate and solve these ratios is essential for finding the length of the shadow of the second tree based on the known measurements of the first.