Measurement Jenny Crum stands on a cliff at the edge of a...

Measurement Jenny Crum stands on a cliff at the edge of a canyon. On the opposite side of the canyon is another cliff equal in height to the one she is on. (See the figure.) By dropping a rock and timing its fall, she determines that it is 105 ft to the bottom of the canyon. She also determines that the angle to the base of the opposite cliff is $27^{\circ} .$ How far is it to the opposite side of the canyon?

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Key Concepts

Trigonometric Functions

Trigonometric functions relate the angles and side lengths of triangles. In measurement problems, the tangent function is particularly useful since it relates the angle to the ratio of the opposite side to the adjacent side, allowing one to calculate unknown horizontal or vertical distances when one of these dimensions and an angle are known.

Right Triangle Geometry

Right triangle geometry is fundamental when modeling situations involving heights and distances. By envisioning a scenario as a right triangle, one can apply trigonometric ratios to solve for missing side lengths, which is essential in problems that involve determining distances across gaps or from a specific height.

Angle of Elevation/Depression

Angles of elevation and depression are measured from the horizontal line to the line of sight, upward for elevation and downward for depression. These angles are critical in forming accurate right triangle models, enabling the use of trigonometric relationships to connect measured angles with actual distance and height values.

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