Problem 55 - Algebra and Trigonometry

Question

When the angle of elevation of the Sun is \(62^{\circ}\), a telephone pole that is tilted at an angle of \(8^{\circ}\) directly away from the Sun casts a shadow 20 feet long. Determine the length of the pole to the nearest tenth of a foot.

Step-by-Step Solution

Answer

To find the length of the telephone pole, calculate the formula \( Height = 20 \times tan(54^{\circ}) \). Round the calculated height to the nearest tenth of a foot.

Step 1: Understand the Problem

The Sun's angle of elevation is the angle between the ground and the line from the object to the Sun. Here the angle of elevation of the Sun is \(62^{\circ}\). The telephone pole is tilted at \(8^{\circ}\) away from the Sun, thus decreases the effective angle of elevation to \(62^{\circ} - 8^{\circ} = 54^{\circ}\). As the problem forms a right triangle, we can apply the tangent formula. Where, tan of the angle = height of the telephone pole (opposite side) / shadow length (adjacent side).

Step 2: Setup the Equation

So, setting up the equation, \[ tan(54^{\circ}) = \frac{Height}{20} \] Now, to find the height, we rearrange the formula to make 'Height' the subject. So, \[ Height = 20 \times tan(54^{\circ}) \]

Step 3: Calculate Height

Then, we calculate the height by plugging the values for shadow length and the angle into the height formula. When we use a scientific calculator, make sure that it's set to degree mode, not radian mode.

Step 4: Round to Nearest Tenth

Finally, the calculated height is rounded to the nearest tenth of a foot as per the requirements of the exercise.

Key Concepts

Angle of Elevation

The angle of elevation is a key term in trigonometry, representing the angle formed between the horizontal line from an observer's eye and the line of sight from the observer to an object above. This term is commonly used when discussing problems involving heights and distances, like the exercise with the Sun and the telephone pole.

Understanding the angle of elevation helps in visualizing how the sun's position affects the shadow cast by an object. For instance, if the Sun is directly overhead, the angle of elevation would be close to 90 degrees, producing a shorter shadow. In our exercise, the angle of elevation is given as 62 degrees. However, due to the tilt of the telephone pole, the effective angle is decreased by the pole's tilt of 8 degrees, resulting in an adjusted angle of 54 degrees.

This adjusted angle is crucial in setting up the correct trigonometric calculations to find the pole's height.

Tangent Formula

The tangent formula is essential when working with right triangles, particularly when you know an angle and a side length, and need to find another side. In trigonometry, the tangent of an angle is the ratio of the opposite side to the adjacent side. This ratio is expressed as:

• \(\tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

In our problem, you'll use the tangent formula to find the height of the telephone pole, which is the opposite side of our angle. The shadow length is the adjacent side, given as 20 feet.

Plugging in the adjusted angle of 54 degrees and the shadow, the formula becomes:

• \(tan(54^{\circ}) = \frac{\text{Height}}{20}\)

By solving this equation, we rearrange to find the height, multiplying both sides by 20:

• \(\text{Height} = 20 \times \tan(54^{\circ})\)

This equation allows us to calculate the height of the pole using the tangent function.

Right Triangle

A right triangle has a right angle, which is an angle of 90 degrees. This is foundational in trigonometry because it allows us to use functions like sine, cosine, and tangent to solve for unknown side lengths or angles. In the context of our exercise, the setup forms a right triangle with the shadow as one side and the pole, in effect, as another.

The right triangle principle simplifies calculations using trigonometric functions for solving real-world problems. The formulae associated with such triangles hinge on understanding the relationships between their angles and sides.

In the given exercise, the right triangle consists of the base (the shadow), the height (the pole), and the hypotenuse (the line from the top of the pole to the tip of the shadow). Recognizing these components helps in correctly applying the tangent formula to calculate the pole's height.

Scientific Calculator

A scientific calculator is a versatile tool that is essential for solving trigonometry problems. It offers functions beyond simple arithmetic, such as calculating trigonometric ratios like sine, cosine, and tangent, which are vital for the exercise.

When calculating the height of the telephone pole, make sure your scientific calculator is set to degree mode. This is important because angles in trigonometry problems are often given in degrees. To set to degree mode, usually, you need to look for a "mode" button and select "degrees". Many calculators default to radians, so this step is crucial.

Once ready, you can input the tangent function needed for the exercise. For example, to find \(\tan(54^{\circ})\), simply type "54", then press the "tan" button. The calculator will provide the correct tangent value, allowing you to complete the equation and find the necessary height of the pole.