Question

You are driving up an inclined road. After \(2.4 \mathrm{~km}\) you notice a roadside sign that indicates that your elevation has increased by \(160 \mathrm{~m}\). What is the angle of the road above the horizontal?

Short answer

The angle of the road above the horizontal is approximately \(3.82^\circ\).

Step-by-step solution

Understand the Problem

To determine the angle of the road above the horizontal, we need to calculate the angle of inclination which can be represented as the angle \( \theta \) in a right triangle, where the hypotenuse is the length of the road (\( 2.4 \) km) and the opposite side is the elevation increase (\( 160 \) m). The trigonometric function which utilizes opposite and hypotenuse is the sine function: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

Convert Units

First, ensure that the units are consistent. The hypotenuse (road length) and the opposite side (elevation increase) should be in the same unit. Convert \( 2.4 \) km to meters: \(2.4 \text{ km} = 2400 \text{ m}\).

Apply the Sine Function

Use the sine function with the values obtained. Substitute into the formula: \( \sin(\theta) = \frac{160}{2400} \).

Calculate the Angle

Calculate \( \sin(\theta) \): \( \sin(\theta) = \frac{160}{2400} = \frac{1}{15} \). To find the angle \( \theta \), take the inverse sine (arcsin) of \( \frac{1}{15} \): \( \theta = \arcsin\left(\frac{1}{15}\right) \). Calculate this using a calculator or trigonometric table to find \( \theta \).

Interpretation

After calculation, \( \theta \approx 3.82^\circ \). This angle represents the incline of the road above the horizontal.

Key concepts

Right Triangle

In the problem scenario of driving up an inclined road, we encounter a practical application of geometric shapes. A right triangle is a specific type of triangle that has one of its angles measuring 90 degrees. This is important because it allows us to use trigonometry to solve various problems, such as finding unknown angles or lengths.

In a right triangle:

• The longest side is called the hypotenuse.
• The other two sides are the opposite side (the one across from the angle in question) and the adjacent side (the one right next to the angle).

For this exercise, the road length (2.4 km or 2400 m once converted) acts as the hypotenuse, while the elevation gain (160 m) serves as the opposite side.
Understanding the structure of a right triangle lets us apply trigonometric functions properly.

Angle of Elevation

In this context, the angle of elevation pertains to the angle formed between the horizontal plane and the line of sight or the inclined plane. This concept serves an essential role in identifying the steepness or slope of a path. The angle of elevation becomes crucial when calculating how much you are climbing on an incline available on slopes or mountainous terrains. In our problem, you find this angle by observing how much vertical distance (160 m) is gained over the slant distance (2400 m).

To visualize:

• Imagine looking straight ahead at a distant plateau. If you need to raise your line of sight to see the top, that adjustment is the angle of elevation.

Through recognizing this angle, you can measure safe travel conditions or adjust the design parameters of roads and pathways.

Sine Function

The sine function is a fundamental element in trigonometry often denoted as \( \sin \theta \). It links a right triangle's angle to the ratio of the lengths of its sides. The formula \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \) illustrates its simplicity and power.

To utilize the sine function in calculations:

• Identify the opposite side relative to the angle \( \theta \): for our problem, it's the elevation gain (160 m).
• Identify the hypotenuse: the total inclined path (2400 m).
• Substitute these values to form the ratio.

Once the ratio is determined, the next step involves figuring out \( \theta \) using the inverse sine or arcsine function. Specifically, you compute \( \arcsin \left( \frac{1}{15} \right) \), yielding an angle of about 3.82 degrees. Applying these calculations helps in understanding the trigonometric functions' applications in real-world scenarios, like road elevation.