Problem 19 - Contemporary Precalculus

Question

In Example \(4,\) suppose that the angle between the two tracks is \(112^{\circ}\) and that the average speeds are 90 kilometers per hour for the first train and 55 kilometers per hour for the second train. How far apart are the trains after two hours and 45 minutes?

Step-by-Step Solution

Answer

Answer: The two trains are approximately 301.7 kilometers apart after 2.75 hours.

Step 1: Convert the time to hours

To make calculations easier, let's convert the given time of "2 hours and 45 minutes" to hours. We can do this by dividing 45 minutes by 60 minutes and then adding this to the 2 hours. 2 hours + (45 minutes / 60 minutes) = 2.75 hours So, the total time interval is 2.75 hours.

Step 2: Find the distance traveled by each train

Now, we'll find the distance traveled by each train after 2.75 hours using the formula distance = speed × time. For the first train: distance1 = 90 km/h × 2.75 h = 247.5 km For the second train: distance2 = 55 km/h × 2.75 h = 151.25 km

Step 3: Use the Law of Cosines to find the distance between the two trains

We now have a triangle with sides (distance1, distance2) and the angle between them (112 degrees). We can use the Law of Cosines to find the distance between the two trains: d^2 = distance1^2 + distance2^2 - 2(distance1)(distance2)cos(112°) where d represents the distance between the trains.

Step 4: Calculate the distance d

Now, we can plug in the values for distance1 and distance2: d^2 = (247.5)^2 + (151.25)^2 - 2(247.5)(151.25)cos(112°) Solve for d: d ≈ 301.7 km

Step 5: Present the final answer

After 2.75 hours, the two trains are approximately 301.7 kilometers apart.

Key Concepts

Distance Calculation

Understanding how to calculate distance is essential for solving a vast array of real-world problems, such as determining how far apart two moving objects like trains will be after a given period. In our exercise, we utilized the fundamental formula for distance, which is given by \( \text{distance} = \text{speed} \times \text{time} \). This formula helps to convert the rate at which an object moves and the duration of its motion into a tangible measure of space covered.

When calculating the distance for each train, one must multiply the speed of the train by the total time traveled. It's important to ensure that the units of speed and time are compatible, a common pitfall for many students. Notice that in the solution provided, both speed (in kilometers per hour) and time (converted into hours) align, allowing for a straightforward multiplication. The distances obtained then act as the sides of a triangle, which will lead to the use of the Law of Cosines for problems involving angles and three-dimensional space.

Trigonometry in Precalculus

Trigonometry, a branch of mathematics dealing with the relationships between the sides and angles of triangles, plays a crucial role in precalculus. One of the most powerful tools within this field is the Law of Cosines. This law allows us to relate the sides of a triangle to the cosine of one of its angles. In situations where the Law of Sines is not applicable, primarily when we deal with oblique triangles where no right angle is present, the Law of Cosines becomes significantly useful.

In the context of our problem, the Law of Cosines formula, written as \( d^2 = a^2 + b^2 - 2ab\cos(\theta) \), was applied, where \(a\) and \(b\) represent the known side lengths (distances traveled by each train), \(\theta\) is the included angle (112 degrees in the case), and \(d\) is the distance between the two trains. Through this trigonometric relationship, the side opposite the known angle can be found, providing the solution to the problem at hand.

Time Conversion

Time conversion often plays a critical role in solving physics and geometry problems, where time needs to be expressed consistently in a single unit. It's common in calculations involving speed and distance to work in hours, given that speeds are frequently given in kilometers or miles per hour.

In our example, the time of '2 hours and 45 minutes' needed to be converted to hours for uniformity with the speed's unit (kilometers per hour). To convert minutes to hours, we divide the minutes by 60, as there are 60 minutes in an hour. Therefore, \( 45 \text{ minutes} \div 60 = 0.75 \text{ hours} \). Adding this to the 2 hours gives us a total of 2.75 hours. By mastering time conversion, students can eliminate potential errors and simplify the process of using formulas in physics and related fields, ensuring accurate calculations.