Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Distance between automobiles Two automobiles leave a city at the same time and travel along straight highways that differ in direction by \(84^{\circ}\). If their speeds are \(60 \mathrm{mi} / \mathrm{hr}\) and \(45 \mathrm{mi} / \mathrm{hr}\), respectively, approximately how far apart are the cars at the end of 20 minutes?

Step-by-Step Solution

Answer

The cars are approximately 23.7 miles apart after 20 minutes.

Step 1: Convert Time to Hours

First, convert the time from minutes to hours to be consistent with the speed units. Since there are 60 minutes in an hour, the time in hours is given by \( \frac{20}{60} = \frac{1}{3} \) hours.

Step 2: Calculate Distances Traveled

Next, calculate the distance each car travels using the formula \( \text{distance} = \text{speed} \times \text{time} \).- For the first car: \( 60 \frac{\text{miles}}{\text{hour}} \times \frac{1}{3} \text{ hour} = 20 \text{ miles} \).- For the second car: \( 45 \frac{\text{miles}}{\text{hour}} \times \frac{1}{3} \text{ hour} = 15 \text{ miles} \).

Step 3: Apply the Law of Cosines

The cars form a triangle with the city as one vertex. The angle between their paths is \( 84^{\circ} \).Apply the law of cosines to find the distance \( d \) between the two cars:\[ d^2 = 20^2 + 15^2 - 2 \times 20 \times 15 \times \cos(84^{\circ}) \].Calculate the cosine of \( 84^{\circ} \) and substitute it into the equation to solve for \( d \).

Step 4: Calculate and Simplify

Now compute:\[ d^2 = 400 + 225 - 600 \times \cos(84^{\circ}) \].Find \( \cos(84^{\circ}) \), approximately 0.1045, and continue:\[ d^2 = 625 - 600 \times 0.1045 \]\[ d^2 = 625 - 62.7 \]\[ d^2 = 562.3 \].Finally, take the square root:\[ d \approx \sqrt{562.3} \approx 23.7 \].

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and lengths of triangles. It plays a crucial role in solving various geometry-related problems. In this exercise, we specifically rely on the Law of Cosines, which is particularly handy when dealing with non-right triangles.

The Law of Cosines helps calculate the unknown side lengths of a triangle when you know either two sides and the included angle or three sides. The formula is an extension of the Pythagorean theorem. While the Pythagorean theorem is limited to right triangles, the Law of Cosines can be applied to any triangle.

The formula for the Law of Cosines is:

• \[ c^2 = a^2 + b^2 - 2ab\cos(C) \]

In this formula, \( a \) and \( b \) are the known sides of the triangle, \( C \) is the angle between them, and \( c \) is the side that's opposite the angle. Understanding this universal trigonometric principle is key in applying it to our problem.

Distance Calculation

Calculating distance is often about understanding the relationship between speed, time, and space. When calculating distances in word problems, we typically use the formula:

• \[ \text{distance} = \text{speed} \times \text{time} \]

This formula is straightforward but requires ensuring consistency in units. For example, if speed is given in miles per hour, time should be converted to hours.

In the original exercise, the cars traveled for 20 minutes. To convert this to hours, divide by 60, resulting in \( \frac{1}{3} \) of an hour. Each car's distance is determined by multiplying its speed by this time period:

• The first car: \( 60 \text{ mph} \times \frac{1}{3} \text{ hr} = 20 \text{ miles} \)
• The second car: \( 45 \text{ mph} \times \frac{1}{3} \text{ hr} = 15 \text{ miles} \)

These distances form the sides of a triangle, where the city is the common starting point.

Triangle Geometry

Triangle geometry involves understanding different types of triangles and how to work with their properties, such as angles and sides. In this problem, the challenge lies in recognizing the type of triangle formed by the cars and utilizing appropriate mathematical tools to find the missing side.

When two objects move away from a common point in different directions, they create a triangle with the point as one vertex and their paths forming the two sides. Here, the angle between the paths is given as \( 84^{\circ} \), and the known side lengths are the distances each car traveled.

This scenario is a typical case for applying the Law of Cosines. Given two sides and an included angle, the Law of Cosines can be used to find the unknown side. After applying this, we end up solving for the distance between the two cars.

Our understanding of triangle properties and the application of the Law of Cosines allows us to determine that the distance between them is approximately 23.7 miles. Familiarity with these concepts of triangle geometry is essential in many real-world problems where direction and distance calculations are necessary.