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?

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Recognize that this problem involves similar triangles formed by the tree and its shadow, where the ratio of the height of the tree to the length of its shadow remains constant at the same time and place.

Set up the ratio using the given information: the height of the first tree (20 ft) over its shadow length (8 ft), which can be written as \(\frac{20}{8}\).

Let the length of the shadow of the 30-ft tree be \(x\). Set up the proportion \(\frac{20}{8} = \frac{30}{x}\) to express the equality of ratios.

Solve the proportion for \(x\) by cross-multiplying: \(20 \times x = 8 \times 30\).

Isolate \(x\) by dividing both sides by 20: \(x = \frac{8 \times 30}{20}\). This expression gives the length of the shadow of the 30-ft tree.

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Similar Triangles

When two triangles have the same angles, they are similar, meaning their corresponding sides are proportional. In this problem, the tree and its shadow form a right triangle, and comparing two such triangles allows us to set up ratios to find unknown lengths.

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30-60-90 Triangles

Proportionality in Right Triangles

The lengths of sides in right triangles that share the same angle maintain a constant ratio. This principle lets us relate the height and shadow length of one tree to another under identical lighting conditions, enabling calculation of unknown shadow lengths.

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30-60-90 Triangles

Application of Ratios

Ratios compare two quantities and are used to solve for unknown values in proportional relationships. Here, the ratio of the height to shadow length of the first tree helps determine the shadow length of the second tree by setting up and solving a proportion.

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