Problem 49 - Algebra and Trigonometry

Question

Distance, Speed, and Time Wendy took a trip from Davenport to Omaha, a distance of 300 mi. She traveled part of the way by bus, which arrived at the train station just in time for Wendy to complete her journey by train. The bus averaged 40 \(\mathrm{mi} / \mathrm{h}\) and the train 60 \(\mathrm{mi} / \mathrm{h}\) . The entire trip took 5\(\frac{1}{2} \mathrm{h} .\) How long did Wendy spend on the train?

Step-by-Step Solution

Answer

Wendy spent 4 hours on the train.

Step 1: Define Variables

Let the time spent on the bus be \( t_1 \) hours. The time spent on the train will then be \( t_2 \) hours. Since the total travel time is given as \( 5.5 \) hours, we have the equation: \[ t_1 + t_2 = 5.5 \]

Step 2: Use Distance Formula for the Bus

The bus travels at \( 40 \) mi/h. If \( t_1 \) is the time taken by the bus, then the distance covered by the bus is \( 40t_1 \) miles. This is expressed by the formula: \[ \text{Distance (Bus)} = 40t_1 \]

Step 3: Use Distance Formula for the Train

The train travels at \( 60 \) mi/h. If \( t_2 \) is the time taken by the train, then the distance covered by the train is \( 60t_2 \) miles. This is given by the formula: \[ \text{Distance (Train)} = 60t_2 \]

Step 4: Set Up Total Distance Equation

The total distance from Davenport to Omaha is \( 300 \) miles. Therefore, the sum of the distance covered by bus and train is equal to \( 300 \): \[ 40t_1 + 60t_2 = 300 \]

Step 5: Solve the System of Equations

We have two equations:1. \( t_1 + t_2 = 5.5 \)2. \( 40t_1 + 60t_2 = 300 \)First, express \( t_1 \) from the first equation: \( t_1 = 5.5 - t_2 \). Substitute \( t_1 \) in the second equation:\[40(5.5 - t_2) + 60t_2 = 300\]

Step 6: Simplify and Solve for t_2

Expand and simplify the equation:\[220 - 40t_2 + 60t_2 = 300\]\[20t_2 = 300 - 220\]\[20t_2 = 80\]Divide both sides by 20:\[t_2 = 4\]

Step 7: Conclusion - Time on Train

Wendy spent \( 4 \) hours on the train, as calculated from solving the equation.

Key Concepts

Systems of Equations

Understanding a systems of equations is integral when dealing with problems of time, speed, and distance. In our example, Wendy's journey was planned so that parts of it took place on a bus, and then on a train. To find out the time she spent on each, we set up a system of two equations.

• The first equation represents the total travel time, given as 5.5 hours.
• The second equation represents the total distance, which is the sum of distances traveled by bus and by train.

By expressing both time and distance in terms of variables, we create two simultaneous equations. The solution involves solving these equations together to find out exactly how long Wendy spent on each mode of transport. Systems of equations allow us to use known information to solve for unknown variables effectively.

Distance Formula

The distance formula is crucial when analyzing problems involving movement. Calculating distance is a straightforward process, but it requires knowledge of speed and time.The formula used is: \[ \text{Distance} = \text{Speed} \times \text{Time} \]In Wendy's scenario, each part of her journey is analyzed separately:

• For the bus part of the journey: The distance is calculated as \(40t_1\) miles, where \(t_1\) is the time spent on the bus and 40 is the speed in miles per hour.
• For the train segment: The distance comes from \(60t_2\) miles, where \(t_2\) is the time on the train, and 60 is the train's speed.

The sum of these distances gives the total journey distance, helping solve for unknowns when combined with other equations.

Travel Time Calculation

Calculating travel time using given speeds and distances helps map out journeys precisely. In this instance, the given total travel time was 5.5 hours.

• First, you determine how much time was spent on each segment using variables \(t_1\) and \(t_2\).
• The equation \(t_1 + t_2 = 5.5\) represents the combined time spent on bus and train.

By substituting expressions for \(t_1\) and \(t_2\) into the distance equations, the total distance equation becomes solvable. This connection enables you to calculate specific travel times for each mode, ultimately narrowing down the itinerary. Here, through systematic calculations, Wendy's train time is revealed as 4 hours, providing the insight needed for accurate travel planning.