Question

Challenge 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 to second base is approximately 36.39 meters.

Step-by-step solution

Understand the Problem

We are given a right triangle where the opposite side (height) is the vertical distance to second base, which is 9.75 meters. The angle given is \(15.0^{\circ}\), which is the angle of depression towards the base. We need to find the horizontal distance (adjacent side) to second base.

Identify the Trigonometric Function to Use

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. Therefore, we will use the tangent function to find the horizontal distance:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\] Here, \(\theta = 15.0^{\circ}\) and the opposite side is 9.75 m. We need to solve for the adjacent side.

Rearrange the Equation

Rearranging the tangent function equation gives us the formula for finding the adjacent side:\[\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}\]Now, substitute the known values into the equation.

Substitute the Values

Insert the values into the rearranged formula:\[\text{adjacent} = \frac{9.75}{\tan(15.0^{\circ})}\]Calculate \(\tan(15.0^{\circ})\) using a calculator.

Calculate the Horizontal Distance

First, calculate \(\tan(15.0^{\circ})\):\(\tan(15.0^{\circ}) \approx 0.2679\).Now use the formula again:\[\text{adjacent} = \frac{9.75}{0.2679} \approx 36.39 \, \text{meters}\]This is the horizontal distance from the press box to the second base.

Key concepts

Angle of Depression

The angle of depression is an interesting and often misunderstood concept in trigonometry. It is defined as the angle formed between the line of sight from an observer to an object below the horizontal line running directly from the observer's eyes.

In simpler terms, it's the angle you look down when you're looking at something below you. For example, if you're standing on a hill looking at a tree at the bottom, the angle of depression would describe that downward glance. Some key things to know about the angle of depression include:

• It's always measured from the horizontal plane.
• It's equal to the angle of elevation from the bottom object to the observer if there's a straight horizontal between them.
• In right triangle problems like the baseball park exercise, it helps to find unknown distances by creating a right triangle with the observer's line of sight as the hypotenuse.

Understanding the angle of depression can make real-life navigation and measurement much simpler in both daily activities and mathematical problems.

Right Triangle

Right triangles are a fundamental part of geometry and trigonometry. They are triangles with one angle measuring exactly 90 degrees. This special property makes them especially useful in various calculations and helps in simplifying many mathematical problems.

Here are a few essential points to remember about right triangles:

• The longest side is known as the hypotenuse, and it is always opposite the right angle.
• The two shorter sides are called legs, and you can use these sides with trigonometric functions to find various angles and lengths.
• Since they follow the Pythagorean theorem: \[a^2 + b^2 = c^2\] where \(c\) is the hypotenuse and \(a\) and \(b\) are the other sides.

In the context of the baseball problem, the problem forms a right triangle with one leg as the vertical distance (height from the press box to the ground) and the other leg as the horizontal distance to the second base. This setup makes it a perfect candidate for using trigonometric functions, like the tangent function, to solve for unknown lengths.

Tangent Function

The tangent function is a core trigonometric function, crucial for working with angles and distances in right triangles. The function relates an angle of a right triangle to the ratio of two of its sides.

In simple terms, if you know one angle of a right triangle and the length of one side, the tangent function can help you find another side. Here's how it works:

• The tangent of an angle \(\theta\) is the ratio of the opposite side to the adjacent side: \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]
• It's useful for finding unknown side lengths when one side and an angle are known.

In the baseball park example, you used the tangent function to calculate the horizontal distance from the press box to second base. Given the opposite side (height) and angle of depression, the tangent function allowed you to rearrange and solve for the unknown adjacent side. This makes the tangent function particularly useful for solving real-world problems involving angles and distances.