Solve each problem. See Example 5. Height of a Building A house is 15 ft tall. Its shadow is 40 ft long at the same time that the shadow of a nearby building is 300 ft long. Find the height of the building.

Verified step by step guidance

1. Identify the right triangles formed by the house and its shadow, and the building and its shadow. Both triangles share the same angle of elevation of the sun, so their corresponding sides are proportional.
2. Set up the proportion using the heights and shadow lengths: the ratio of the height of the house to its shadow length equals the ratio of the height of the building to its shadow length. Mathematically, this is \(\frac{15}{40} = \frac{h}{300}\), where \(h\) is the height of the building.
3. Cross-multiply to solve for \(h\): multiply 15 by 300 and 40 by \(h\), giving \(15 \times 300 = 40 \times h\).
4. Isolate \(h\) by dividing both sides of the equation by 40: \(h = \frac{15 \times 300}{40}\).
5. Simplify the expression to find the height of the building.

Key Concepts

Similar Triangles

When two objects cast shadows at the same time under the same light source, the triangles formed by their heights and shadows are similar. This means their corresponding sides are proportional, allowing us to set up ratios to find unknown lengths.

Proportionality in Right Triangles

In right triangles formed by the height and shadow of an object, the ratio of height to shadow length remains constant for objects under the same lighting conditions. This proportionality helps solve for unknown heights or shadow lengths.

Setting Up and Solving Ratios

To find the unknown height, we create a ratio comparing the known height and shadow length of the house to the unknown height and shadow length of the building. Solving this proportion involves cross-multiplication and basic algebra.