Problem 76 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. The angle of elevation of the top of a hill with respect to a certain point on the surrounding level ground is \(10^{\circ} .\) If the hill is 9 feet high, what is the horizontal distance of the top of the hill from that point?

Step-by-Step Solution

Answer

The horizontal distance of the top of the hill from the point of observation is approximately 51.4594 feet.

Step 1: Recognize the right triangle and symbols

Recognize that the problem involves a right triangle where the height of the hill is the opposite side (O), the horizontal distance is the adjacent side (A), and the angle of elevation is the angle between these sides (\(\theta\)). So, \(\tan(\theta) = \frac{O}{A}\).

Step 2: Substitute given values

Substitute the given values into the equation. The angle \(\theta = 10^\circ\) and the opposite (O) is 9 feet. The equation becomes: \(\tan(10^\circ) = \frac{9}{A}\).

Step 3: Solve for the horizontal distance

Rearrange the equation to solve for A: \(A = \frac{9}{\tan(10^\circ)}\). Calculate the tangent of the angle to get a numeric value. After performing the calculation, round the result to four decimal places as stated in the problem.

Key Concepts

Understanding Right Triangles

A right triangle is a special kind of triangle that has one angle measuring 90 degrees. This angle is also called a right angle. The right triangle is often used in trigonometry because its properties are easy to understand and apply in solving problems. The triangle has three sides: the hypotenuse, the opposite, and the adjacent.

• The hypotenuse is the side opposite the right angle and is the longest side of the triangle.
• The opposite side is the side opposite to the angle of interest other than the right angle.
• The adjacent side is the side next to the angle of interest besides the hypotenuse.

In problems involving right triangles, such as the one about the hill, it's essential to correctly identify these sides. Knowing which side is which helps determine the right trigonometric ratios to use. This understanding makes solving real-world problems like calculating distances or heights much easier.

The Angle of Elevation and Its Role

The angle of elevation is the angle formed by the line of sight and the horizontal plane when looking up at an object. This angle is a critical concept when dealing with right triangles in real-life scenarios such as architecture or navigation.
To visualize this, imagine standing at a point looking up at the top of a hill. The line from your eyes to the top of the hill creates an angle with the horizontal ground where you stand. This angle is the angle of elevation.

• It helps determine relationships between different sides of a right triangle.
• Through trigonometric functions like tangent, it helps solve for unknown distances or heights.

In the exercise above, the angle of elevation is given as 10 degrees, helping define the relationship between the height of the hill (opposite side) and the horizontal distance (adjacent side). Understanding and identifying this angle can significantly simplify solving related trigonometry problems.

Using the Tangent Function

The tangent function, often written as `tan`, is a trigonometric function that relates the angles and sides of a right triangle. It is specifically useful in scenarios where the angle and at least one side length of a right triangle are known, as it allows for the calculation of other sides.
For an angle \( \theta \) in a right triangle, the tangent function is defined as the ratio of the opposite side to the adjacent side: \[ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \]
In the given problem, you use \( \tan(10^\circ) = \frac{9}{A} \) to find the horizontal distance. Here:

• Opposite side (O) is the height of the hill, given as 9 feet.
• Adjacent side (A) is the horizontal distance which we solve for.

By rearranging the equation, \( A = \frac{9}{\tan(10^\circ)} \), you can find the required distance. Calculating often involves approximating \( \tan(10^\circ) \) using a calculator, ensuring the result is rounded to four decimal places as required.