Question

Finding the Distance to a Plateau Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is \(60^{\circ},\) how far are you from the base of the plateau?

Short answer

Approximately 28.87 meters

Step-by-step solution

- Identify known values

The height of the plateau is given as 50 meters and the angle of elevation to the top of the plateau is given as \(60^{\circ}\).

- Set up the trigonometric function

To find the distance to the base, use the tangent function in trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]Here, \(\theta = 60^{\circ}\), the opposite side is 50 meters (the height), and the adjacent side is the distance we need to find.

- Write the equation

Substitute the known values into the tangent equation:\[\tan(60^{\circ}) = \frac{50}{d}\]where \(d\) is the distance to the base.

- Solve for distance

Recall that \(\tan(60^{\circ}) = \sqrt{3}\). Now, rewrite the equation:\[\sqrt{3} = \frac{50}{d}\]To solve for \(d\), multiply both sides by \(d\) and then divide both sides by \sqrt{3}:\[d = \frac{50}{\sqrt{3}} = \frac{50 \sqrt{3}}{3}\]

- Simplify the expression (if needed)

The distance can be further simplified if necessary. Here, \(d\) becomes \[\approx 28.87 \text{ meters}\]

Key concepts

Angle of Elevation

The angle of elevation is a key concept in trigonometry and helps us solve real-world problems involving heights and distances. Imagine standing on the ground, looking up at the top of a plateau. The angle your line of sight makes with the horizontal ground is the angle of elevation. In this exercise, the angle of elevation is given as 60 degrees.

To understand it better, let's break it down:

• Angle of elevation is always measured from a horizontal line upwards to the line of sight.
  
• This angle is crucial for setting up trigonometric functions, as it gives us a clear path to solve for unknown distances (like the distance to the base of the plateau).
  

When dealing with problems like these, always identify the angle of elevation first. It guides you to choose the right trigonometric function (in this case, the tangent function). Without knowing the angle of elevation, we wouldn't be able to proceed further.

Tangent Function

The tangent function is a central trigonometric function used extensively in various applications, especially when dealing with right triangles. It is particularly useful when one side of the triangle is known (either height or base) and you need to find the other side.

Here's the formula for the tangent function in a right triangle: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

To use the tangent function, we need:

• The angle in question (θ), which in this exercise is 60 degrees.
  
• The lengths of the opposite side (here, the height of the plateau, 50 meters).
  
• The adjacent side (the distance from the observer to the base), which we are solving for.
  

Once we substitute the values into the formula, we can rearrange it to solve for the unknown distance. By knowing that \[\tan(60^{\text{circ}}) = \sqrt{3}\], we quickly solve for the distance. Remember, the tangent function is ideal for these types of problems because it directly relates an angle to two sides of a right triangle.

Right Triangle

Understanding the structure of a right triangle is fundamental in trigonometry. A right triangle contains one 90-degree angle, making it unique and very useful for applying trigonometric functions. In our problem, the right triangle formed by the height of the plateau, the distance to the base, and the line of sight where the observer stands is crucial.

Here's how a right triangle is typically laid out:

• One angle is always 90 degrees.
  
• The side opposite the 90-degree angle is called the hypotenuse.
  
• The other two sides are known as the adjacent (next to the angle) and the opposite (across from the angle) sides.
  

In this exercise:

• The height of the plateau is the opposite side (50 meters).
  
• The distance to the base is the adjacent side.
  
• The line of sight (if measured) would be the hypotenuse, but it's not needed for solving this problem.
  

By identifying these sides and understanding their relationships, we simplify solving the problem. The right triangle allows us to apply the tangent function effectively, leading us directly to the solution.