Problem 80 - Intermediate Algebra

Question

A 62-foot guy-wire makes an angle of \(60^{\circ}\) with the ground and is attached to a telephone pole (see Figure 6.6). Find the distance from the base of the pole to the point on the pole where the wire is attached. Express your answer to the nearest tenth of a foot.

Step-by-Step Solution

Answer

The distance is approximately 53.7 feet.

Step 1: Understand the Problem

You are given a guy-wire of 62 feet long making an angle of \(60^{\circ}\) with the ground. You need to find the vertical distance from the ground (base of the pole) to where the wire is attached to the pole.

Step 2: Identify Relevant Trigonometric Function

The wire, pole, and the ground form a right-angled triangle. To find the vertical height of the pole, use the sine function, as sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.

Step 3: Set Up the Equation

Let \(h\) be the vertical distance. The equation using the sine function is: \( \sin(60^{\circ}) = \frac{h}{62} \).

Step 4: Calculate \(\sin(60^{\circ})\)

The value of \(\sin(60^{\circ})\) is \(\frac{\sqrt{3}}{2}\) or approximately 0.866.

Step 5: Solve for \(h\)

Substitute the value of \(\sin(60^{\circ})\) into the equation and solve for \(h\): \( 0.866 \times 62 = h \).

Step 6: Calculate Exact Height

Multiply and round the result to the nearest tenth: \( h = 53.692 \), thus \( h \approx 53.7 \) feet.

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. In a right triangle, the sides are given special names. The longest side, opposite the right angle, is called the hypotenuse. The other two sides, which form the right angle, are called the legs. These are the adjacent and opposite sides in relation to an angle.
An important property of right triangles is that they obey the Pythagorean theorem. This theorem states that the square of the hypotenuse length is equal to the sum of the squares of the other two sides. Therefore, if you know the lengths of two sides, you can easily find the third. Right triangles are foundational in trigonometry because they allow for simple calculation of angles and lengths using trigonometric functions.

Sine Function

The sine function is an important trigonometric function that relates an angle in a right triangle to a ratio of two sides. Specifically, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. This is expressed as:

• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

The sine function is very useful for finding missing side lengths or angles in a right triangle. Knowing the angle and the hypotenuse allows us to find the length of the opposite side. This capability is particularly helpful in problems like the guy-wire scenario, where we needed to find the pole's height using the sine of a 60-degree angle.

Angle of Elevation

The angle of elevation is the angle formed by the horizontal line of sight and the line of sight up to an object. This angle is always measured from the ground upwards. In practical terms, if you stand and look up at an object like a hill or a tall tower, the angle your eyes make with the ground is the angle of elevation.
In our exercise, the angle of elevation was given as 60 degrees. This angle is crucial because it helps set up the correct trigonometric relationships in our right triangle. Using angles of elevation, you can solve many real-world problems that involve height and distances.

Trigonometric Ratios

Trigonometric ratios are fundamental in studying relationships between the angles and sides of right triangles. There are three primary trigonometric ratios: sine, cosine, and tangent, each associated with specific side ratios of a right triangle.
For the sine function, which we used in this exercise, the ratio is between the opposite side length and the hypotenuse. Meanwhile, cosine relates the adjacent side to the hypotenuse, and tangent uses the opposite over the adjacent side. These ratios can be remembered through the mnemonic "SOHCAHTOA":

• Sine = Opposite/Hypotenuse
• Cosine = Adjacent/Hypotenuse
• Tangent = Opposite/Adjacent

Using these ratios allows us to solve for unknown sides or angles, making them powerful tools in trigonometry for both theoretical and practical applications, like calculating the height of a pole using a guy-wire measurement.