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Chapter 2: Problem 8

Two trains leave the same train station at the same time, moving along straight tracks that form a \(35^{\circ}\) angle. If one train travels at an average speed of \(100 \mathrm{mi} / \mathrm{hr}\) and the other at an average speed of \(90 \mathrm{mi} / \mathrm{hr}\), how far apart are the trains after half an hour?

Short Answer

Expert verified
The trains are about 29 miles apart after half an hour.

Step by step solution

01

Understand the Problem

The problem involves two trains leaving the station at the same time and traveling at different speeds along tracks that form a 35掳 angle. The goal is to find the distance between the trains after half an hour.
02

Calculate Distances Traveled

The first train travels at 100 mi/hr for 0.5 hours. This means it travels: oFirst_train_distance = speed \times time \[100 \text{ mi/hr} \times 0.5 \text{ hr} = 50 \text{ miles} \] The second train travels at 90 mi/hr for 0.5 hours. This means it travels: oSecond_train_distance = speed \times time \[90 \text{ mi/hr} \times 0.5 \text{ hr} = 45 \text{ miles} \]
03

Use the Law of Cosines

We need to find the distance between the two trains using the Law of Cosines. The formula is: \[ c^2 = a^2 + b^2 - 2ab \times \text{cos}(\theta) \]where: (a and b are the distances traveled by each train, and 胃 is the angle between them. Inserting our values: a = 50 \text{ miles} b = 45 \text{ miles} theta = 35^\text{掳}Then: \[ c = \text{鈭殅(50^2 + 45^2 - 2 \times 50 \times 45 \times \text{cos}(35^\text{掳})) \]
04

Calculate

Using the values in the formula: \[ c = \text{鈭殅(2500 + 2025 - 4500 \times \text{cos}(35^\text{掳})) \] Using \text{cos}(35^\text{掳}) 鈮� 0.8192:\[ c = \text{鈭殅(2500 + 2025 - 4500 \times 0.8192) \] \[ c = \text{鈭殅(2500 + 2025 - 3686.4) \] \[ c = \text{鈭殅(838.6) 鈮� 28.97(or ~29 miles)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Law of Cosines
The Law of Cosines is crucial for solving problems involving triangles, especially when you know two sides and the included angle. It's a generalized version of the Pythagorean theorem, used for any triangle, not just right-angled ones. The formula is \( c^2 = a^2 + b^2 - 2ab \times \text{cos}(\theta) \). In the provided exercise, \(a\) and \(b\) are the distances traveled by the trains and \(\theta\) is the angle between their paths. When applied correctly, it helps us find the distance (\(c\)) between the two trains.

For better understanding:
  • Use this formula when you know two side lengths and the included angle.
  • The angle must be between the two known sides.
  • Substitute the known values correctly into the formula.
  • Use a calculator for trigonometric functions and square roots.
Angle between Vectors
Understanding the angle between vectors is essential in trigonometry. Here, the vectors represent the trains' paths, forming an angle of \(35^{\text{掳}}\). When solving such problems, always ensure to convert the angle to radians if needed (usually for complex calculations). Trigonometric functions in calculators often assume radian mode by default, but for this problem, degrees work fine.

Tips for clarity:
  • Angles in vector-related problems are often measured in degrees; check the problem requirements.
  • Be mindful of the angle's location鈥攊t should be between the vectors (or paths) you are considering.
  • In more advanced applications, knowing how to convert angles (degrees to radians and vice versa) is beneficial.
Distance Calculation
Calculating distance is about applying the right formulas and understanding the context. In this exercise, after using the speeds and time to find the traveled distances (- train 1: 50 miles, train 2: 45 miles), we worked towards finding the separation distance. This distance was calculated with the Law of Cosines due to the angle (not being 90掳) between them.

Key steps include:
  • Breaking down the problem: Find distances first (\( \text{Distance = Speed} \times \text{Time} \)).
  • Using the correct formula (here, Law of Cosines) for more details.
  • Substitute with care: Ensure all calculations align with physical distances and angles.
  • Double-check units鈥攅nsure speed, distance, and time units are consistent.
Trigonometric Functions
Trigonometric functions like cosine, sine, and tangent help solve problems involving angles and sides of triangles. In this problem, cosine (\( \text{cos}(35^\text{掳}) \)) is used. These functions relate angles to side lengths. Using a calculator, you can find trigonometric values, essential for accurate results.

Remember:
  • Basic functions include sine (\( \text{sin} \)), cosine (\( \text{cos} \)), and tangent (\( \text{tan} \)).
  • Sine relates the opposite side to hypotenuse, cosine relates adjacent side to hypotenuse, and tangent involves the opposite to adjacent ratio.
  • Trigonometric values might need a calculator鈥攅nsure knowing how to switch between modes (degree/radian).
In sum, trigonometry builds the path for solving complex real-world distance problems effectively.

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Most popular questions from this chapter

Let \(\triangle A B C\) be an equilateral triangle whose sides are of length \(a\). (a) Find the exact value of the radius \(R\) of the circumscribed circle of \(\triangle A B C\). (b) Find the exact value of the radius \(r\) of the inscribed circle of \(\triangle A B C\). (c) How much larger is \(R\) than \(r\) ? (d) Show that the circumscribed and inscribed circles of \(\triangle A B C\) have the same center.

Show that for any triangle \(\triangle A B C\), \(\cos A+\cos B+\cos C=\frac{a^{2}(b+c-a)+b^{2}(a+c-b)+c^{2}(a+b-c)}{2 a b c} .\)

Draw the triangle \(\triangle A B C\) and its circumscribed and inscribed circles accurately, using a ruler and compass (or computer software). \(a=5\) in, \(b=6\) in, \(c=7\) in

For any triangle \(\triangle A B C\), show that \(c=b \cos A+a \cos B\). This is another check of a triangle.

Show that the triangle area formula $$\text { Area }=K=\frac{1}{4} \sqrt{(a+(b+c))(c-(a-b))(c+(a-b))(a+(b-c))}$$ is equivalent to Heron's formula. (Hint: In Heron's formula replace \(s\) by \(\frac{1}{2}(a+b+c) .\).)
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