Problem 24 - Algebra 2 and Trigonometry

Question

Grace walked within range of a cell phone tower. As soon as her cell phone received a signal, she looked up at the tower. The cotangent of the angle of elevation of the top of the tower is \(\frac{1}{10} .\) If the top of the tower is 75 feet above the ground, to the nearest foot, how far is she from the cell phone tower?

Step-by-Step Solution

Answer

Grace is 8 feet from the cell phone tower.

Step 1: Understanding the Problem

We need to find the horizontal distance Grace is from the cell phone tower. Given the cotangent of the angle of elevation is \( \frac{1}{10} \), and the height of the tower is 75 feet.

Step 2: Relating Cotangent to Distance

The cotangent of an angle in a right triangle is the ratio of the adjacent side (distance from the tower) to the opposite side (height of the tower). This can be expressed as \( \cot(\theta) = \frac{d}{75} = \frac{1}{10} \), where \( d \) is the distance from the tower.

Step 3: Solving for Distance

Rearrange the equation for \( d \):\[ d = 75 \times \frac{1}{10}\]Calculate \( d \):\[ d = 7.5\]Therefore, Grace is approximately 7.5 feet from the tower.

Step 4: Rounding the Final Answer

Since the problem asks for the distance to the nearest foot, round 7.5 to the nearest whole number. Hence, Grace is 8 feet from the cell phone tower.

Key Concepts

Right Triangle

A right triangle is a type of triangle that features a 90-degree angle. This means one of the three angles is always a right angle. In the context of our problem, the right triangle forms between the tower, Grace, and the ground. The sides of this triangle are the height of the tower (the opposite side), the distance from Grace to the tower (the adjacent side), and a hypotenuse that isn't needed here.
Right triangles are useful in trigonometric problems because they allow us to easily apply trigonometric ratios such as sine, cosine, and cotangent. Understanding how to visualize and draw the right triangle, using the tower as the height and the ground as the base, is crucial in solving problems like Grace's distance to the tower.

Angle of Elevation

The angle of elevation is defined as the angle formed between the horizontal line and the line of sight looking up at an object. In this problem, Grace looks up at the top of the tower, creating her angle of elevation.
When dealing with the angle of elevation, it's important to understand that this angle is measured from the ground upward. Calculating the angle can involve trigonometric functions, but in our scenario, we rely on given cotangent values rather than measuring the angle directly. This angle provides us the means to use trigonometric ratios effectively to determine distances.

Trigonometric Ratios

Trigonometric ratios are vital in understanding relationships within right triangles. There are six main ratios: sine, cosine, tangent, cosecant, secant, and cotangent. For Grace's problem, the focus is on the cotangent. Cotangent is the reciprocal of tangent and is defined as the ratio of the adjacent side over the opposite side in a right triangle.
When we know the cotangent of an angle and one side of the triangle, we can find the other side. In Grace's case, with a cotangent of \(\frac{1}{10}\) and a tower height of 75 feet, we solve for the adjacent side. Remembering that \(\cot(\theta) = \frac{adjacent}{opposite}\), or \(\frac{d}{75} = \frac{1}{10}\), helped us calculate the distance effectively. This demonstrates how understanding and applying the right trigonometric ratio can simplify finding unknown measures in geometric problems.