Question

(a) If the angle of the chair lift is decreased, will the horizontal distance \(d_{x}\) increase, decrease, or stay the same? Assume that the length of the lift remains the same, \(d=190 \mathrm{~m}\). (b) Find \(d_{x}\) for the angle \(\theta=15^{\circ}\).

Short answer

(a) Increase; (b) 183.52 m

Step-by-step solution

Understanding the Problem

The problem is about finding how the horizontal distance, \(d_x\), changes with the angle, \(\theta\), when the length of the lift, \(d\), remains constant at 190m. Also, we need to calculate \(d_x\) for a given angle of \(\theta = 15^{\circ}\).

Relationship Between Angle and Distance

The horizontal distance, \(d_x\), can be expressed using trigonometry as \(d_x = d \cdot \cos(\theta)\). When \(\theta\) decreases, \(\cos(\theta)\) becomes larger because the cosine function increases as the angle decreases (from 90 degrees down to 0 degrees). Thus, \(d_x\) will increase if \(\theta\) decreases.

Calculate Horizontal Distance for Given Angle

Given \(\theta = 15^{\circ}\) and \(d = 190\, \mathrm{m}\), we can calculate \(d_x\) as:\[ d_x = 190 \cdot \cos(15^{\circ}) \]Substitute the value of \(\cos(15^{\circ}) \approx 0.9659\) into the equation:\[ d_x = 190 \cdot 0.9659 = 183.521 \mathrm{~m} \]

Conclusion

With a decrease in the angle \(\theta\), the horizontal distance \(d_x\) increases because \(\cos(\theta)\) increases. For an angle of \(15^{\circ}\), the horizontal distance \(d_x\) is approximately \(183.52 \mathrm{~m}\).

Key concepts

Cosine Function

The cosine function relates an angle of a right triangle to the ratio of the adjacent side over the hypotenuse. It's represented as:

• \( \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

This function is one of the primary trigonometric functions, and it's especially useful in various applications, including physics and engineering.
When dealing with angles that range from 0° to 90°, the cosine function produces values from 1 to 0. This means:

• As the angle 🚀 decreases from 90° to 0°, \( \cos(\theta) \) moves closer to 1. This indicates that the adjacent side becomes longer relative to the hypotenuse.

It's important to note that the cosine function accurately describes changes in horizontal distances in practical scenarios. When you know two sides of a right triangle and the angle between them, you can find the third side using the cosine function. This is exactly what we did in our problem when determining the horizontal distance \( d_x \).

Angle of Elevation

The angle of elevation is often used in trigonometry to determine heights and distances that are not easily measured directly. It's the angle formed by the horizontal line of sight and the line joining the observer with an object that is above the horizontal line.
In practical terms, consider looking up at a chair lift or a mountain. The angle at which you tilt your head upwards from horizontal is what we call the angle of elevation. It's the opposite of the angle of depression, which involves looking down.

• When an object, like a chair lift, is at a fixed length and the angle of elevation decreases, it implies that the horizontal component or distance increases, based on the cosine relationship.

The angle of elevation is not only useful for geometric calculations but also serves as a critical component in solving real-world issues involving height and distance.

Horizontal Distance

Horizontal distance is the direct distance measured on the horizontal plane. In the context of trigonometry and this problem, it refers to how far away an object is on the horizontal line of sight from the observer or a given point.
This concept becomes crucial when working with inclined or elevated objects. In scenarios where there's a length at a fixed inclination, changes in the angle of elevation affect horizontal distance.

• When the angle of elevation \( \theta \) decreases, the horizontal distance \( d_x \) increases due to the cosine of the angle increasing.

For example, when the length of the lift is unchanged, any decrease in the angle of elevation translates to a more extended horizontal reach. It's an essential calculation to ensure proper understanding and planning in construction, navigation, and even sports where distances must be measured or evaluated correctly.