Solve each problem. See Examples 3 and 4. Length of a Shadow Suppose that the angle of elevation of the sun is 23.4°. Find the length of the shadow cast by a person who is 5.75 ft tall.
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Verified step by step guidance

1. Identify the right triangle formed by the person, the shadow, and the line of sight to the sun. The person's height is the side opposite the angle of elevation, and the shadow length is the adjacent side.
2. Label the angle of elevation as \(\theta = 23.4^\circ\), the height of the person as the opposite side \(O = 5.75\) ft, and the shadow length as the adjacent side \(A\) (which we need to find).
3. Use the tangent trigonometric ratio, which relates the opposite side and adjacent side in a right triangle: \(\tan(\theta) = \frac{O}{A}\).
4. Rearrange the formula to solve for the shadow length \(A\): \(A = \frac{O}{\tan(\theta)}\).
5. Substitute the known values into the equation: \(A = \frac{5.75}{\tan(23.4^\circ)}\), and then calculate the value of \(A\) to find the length of the shadow.

Key Concepts

Angle of Elevation

The angle of elevation is the angle formed between the horizontal ground and the line of sight to an object above the horizontal. In this problem, it represents the sun's position relative to the person, which helps determine the shadow length using trigonometric ratios.

Right Triangle Trigonometry

The situation forms a right triangle where the person's height is one leg, the shadow length is the other leg, and the angle of elevation is between the shadow and the line to the sun. Using trigonometric functions like tangent relates these sides and the angle.

Tangent Function

Tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, tan(23.4°) = height of person / length of shadow, allowing calculation of the shadow length by rearranging the formula.