Question

The press box at a baseball park is \(9.75 \mathrm{~m}\) above the ground. A reporter in the press box looks at an angle of \(15.0^{\circ}\) below the horizontal to see second base. What is the horizontal distance from the press box to second base?

Short answer

The horizontal distance is approximately 36.39 meters.

Step-by-step solution

Define the Problem

We have a right triangle formed by the press box, second base, and the ground. The distance from the press box directly down to the ground is the vertical side of the triangle, which is 9.75 m. The angle of depression from the horizontal to second base is 15.0°.

Identify the Trigonometric Function

The problem requires finding the horizontal distance from the press box to second base. We will use the tangent function because it relates an angle to the opposite side (vertical distance) and the adjacent side (horizontal distance) in a right triangle.The formula is: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\] where \(\theta = 15.0^{\circ}\).

Rearrange the Equation

To find the horizontal distance (the adjacent side of the triangle), rearrange the tangent formula:\[\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}\]

Substitute Values

Substitute the given values into the equation:\[\text{opposite} = 9.75 \, \mathrm{m}\]\[\theta = 15.0^{\circ}\]Now we calculate:\[\text{adjacent} = \frac{9.75 \, \mathrm{m}}{\tan(15.0^{\circ})}\]

Calculate the Horizontal Distance

Use a calculator to find \(\tan(15.0^{\circ})\), which is approximately 0.2679. Plug this into the equation to find the horizontal distance:\[\text{adjacent} \approx \frac{9.75}{0.2679} \approx 36.39 \, \mathrm{m}\]

Conclude the Solution

Therefore, the horizontal distance from the press box to second base is approximately 36.39 meters.

Key concepts

Angle of Depression

The angle of depression is a key concept in trigonometry, especially when dealing with heights and distances from a horizontal line. Imagine you're standing on a hill looking down at a specific point below. The angle of depression is the angle formed between your line of sight when looking down and the horizontal line above it.

This concept is particularly important when calculating distances or heights using trigonometric ratios, as it allows for a right angle formation necessary for trigonometric analysis.

• The angle is always measured downwards from the horizontal.
• It is equal to the angle of elevation from the point observed back up to the horizontal plane due to alternate interior angles formed with parallel lines.

In this exercise, the reporter's angle of depression from the press box was 15.0°, helping us find the horizontal distance to the second base.

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This feature distinguishes right triangles and is crucial for using trigonometric functions. In any right triangle, the side opposite the right angle is the longest and is called the hypotenuse.

Right triangles are widely used in mathematics and real-life applications because they allow us to apply trigonometric ratios such as sine, cosine, and tangent to solve problems involving angles and lengths of sides.

• They consist of two legs and a hypotenuse.
• The two non-right angles in the triangle always add up to 90 degrees.

In the problem, the vertical line (9.75 m from the press box to ground) and the horizontal line form a right triangle, which we utilized to calculate the position of second base using trigonometry.

Tangent Function

The tangent function is one of the basic trigonometric functions that relate an angle of a right triangle to the ratio of the opposite side to the adjacent side. It's paramount for solving problems involving indirect measurement.

Tangent is especially helpful when direct measurements of distances are complex or impossible. It's usually denoted as \( \tan(\theta) \), where \( \theta \) is the angle of interest.

• Useful for finding the length of one side when the other side's length and the angle are known.
• It involves no hypotenuse, focusing instead on the relationship between the other two sides.

In the exercise given, the formula \[ \tan(15.0^\circ) = \frac{\text{opposite}}{\text{adjacent}} \] was used to determine the unknown distance to second base, effectively translating our angle of depression into practical measurement.