Ch. 1 - Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 1 - Functions

Problema 69

Capitolo 1, Problema 69

Two wires stretch from the top T of a vertical pole to points B and C on the ground, where C is 10 m closer to the base of the pole than is B. If wire BT makes an angle of 35° with the horizontal and wire CT makes an angle of 50° with the horizontal, how high is the pole?

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Identify the right triangles formed by the wires and the pole. Let the height of the pole be h, the distance from the base of the pole to point B be x, and the distance from the base of the pole to point C be x - 10 m.

Use trigonometry to express the height of the pole in terms of x. For triangle T-B, use the tangent function: \( \tan(35^\circ) = \frac{h}{x} \). This gives the equation: \( h = x \cdot \tan(35^\circ) \).

Similarly, for triangle T-C, use the tangent function: \( \tan(50^\circ) = \frac{h}{x - 10} \). This gives the equation: \( h = (x - 10) \cdot \tan(50^\circ) \).

Set the two expressions for h equal to each other: \( x \cdot \tan(35^\circ) = (x - 10) \cdot \tan(50^\circ) \).

Solve the equation for x, and then substitute back into either expression for h to find the height of the pole.

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Trigonometric Ratios

Trigonometric ratios relate the angles and sides of a right triangle. In this problem, the sine, cosine, and tangent functions will be used to find the height of the pole based on the angles formed by the wires with the horizontal. For instance, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, which is crucial for determining the height of the pole.

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Percorso guidato

6:04

Introduction to Trigonometric Functions

Right Triangle Properties

The problem involves right triangles formed by the vertical pole and the wires. Each wire creates a right triangle with the pole as one side and the horizontal distance to points B and C as the other side. Understanding the properties of right triangles, including the Pythagorean theorem, is essential for calculating the height of the pole based on the lengths of the sides and the angles given.

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Percorso guidato

06:21

Properties of Functions

Angle of Elevation

The angle of elevation is the angle formed between the horizontal line and the line of sight to an object above the horizontal. In this scenario, the angles of elevation from points B and C to the top of the pole are given, which allows us to set up equations using trigonometric functions to find the height of the pole. Recognizing how these angles relate to the height and distances involved is key to solving the problem.

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Trig Values in Quadrants II, III, & IV Example 2

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