Problem 35 - Intermediate Algebra

Question

Rates of Speed. Two trains made the same 315 -mile run. since one train traveled 10 mph faster than the other, it arrived 2 hours earlier. Find the speed of each train.

Step-by-Step Solution

Answer

The slower train travels at 35 mph and the faster train at 45 mph.

Step 1: Define Variables and Equations

Let the speed of the slower train be \( x \) mph. Consequently, the speed of the faster train will be \( x + 10 \) mph, because it was said to travel 10 mph faster.

Step 2: Create Time Relations

The time taken by the slower train to cover 315 miles is \( \frac{315}{x} \) hours. Similarly, the time taken by the faster train is \( \frac{315}{x+10} \) hours.

Step 3: Set Up the Equation

Since the faster train arrives 2 hours earlier than the slower one, the equation relating their travel times is:\[ \frac{315}{x} = \frac{315}{x+10} + 2 \]

Step 4: Solve the Equation

Multiply all terms by \( x(x+10) \) to eliminate the denominators:\[ 315(x + 10) = 315x + 2x(x + 10) \]Simplify the equation:\[ 315x + 3150 = 315x + 2x^2 + 20x \]Cancel out \( 315x \) from both sides and rearrange the terms:\[ 2x^2 + 20x - 3150 = 0 \]

Step 5: Factor the Quadratic Equation

Factor the quadratic equation:\( 2x^2 + 20x - 3150 = 0 \).Divide the entire equation by 2:\( x^2 + 10x - 1575 = 0 \).Factor further:\( (x - 35)(x + 45) = 0 \).

Step 6: Determine Possible Solutions

The solutions to the equation are \( x = 35 \) or \( x = -45 \). Since a speed cannot be negative, the slower train's speed is 35 mph. The faster train's speed is \( 35 + 10 = 45 \) mph.

Key Concepts

Train Speed Problems

Train speed problems often involve scenarios where two trains are traveling at different speeds and times. In this exercise, both trains are running the same route of 315 miles. What's interesting is that one train is 10 mph faster yet takes 2 hours less than the slower one to reach the destination. Such problems require a careful setup of equations based on the given details.

• Two trains moving along the same track.
• One train moving faster by a set amount (10 mph faster).
• The challenge of figuring out each train's speed.

These scenarios are practical problems that can be solved by understanding the relation between speed, distance, and time.

Time-Distance Relationship

In a train speed problem, the time-distance relationship is crucial. Distance is the product of speed and time. For our problem:

• The slower train covers 315 miles in a time frame defined by its speed.
• The faster train covers the same 315 miles, but more quickly, due to its higher speed.

Let's consider the formula used: The time taken by the slower train is given by:\[ t = \frac{315}{x} \] For the faster train, it's:\[ t = \frac{315}{x+10} \] This relationship is vital for setting up the equation that compares the two time durations. Understanding how to manipulate and use these formulas can help solve similar problems.

Solving Quadratic Equations

The heart of this problem involves solving a quadratic equation, a fundamental mathematical skill. Quadratic equations can be recognized by their standard formula form:\[ ax^2 + bx + c = 0 \] In the exercise, the equation derived was:\[ 2x^2 + 20x - 3150 = 0 \] Simplifying, we present it as: \[ x^2 + 10x - 1575 = 0 \] We used factoring to find the solutions. Factoring involves breaking down the quadratic into a product of two binomials:\[ (x - 35)(x + 45) = 0 \] Solving these equations provided potential speeds: 35 and -45. Since speed cannot be negative, we choose 35 mph for the slower train, confirming 45 mph for the faster train. Solving quadratics helps us find such real-world unknowns efficiently.

Rate Problems

Rate problems like this one focus on understanding how different speeds affect travel times when distances remain constant. These problems illustrate how minor differences in speed can result in significant time differences over long distances.

• The equation highlights how the rate of speed affects timing.
• Slower speeds result in longer travel times.
• Faster speeds shorten travel times.

In the context of this exercise, recognizing how the faster train's speed advantage results in arriving sooner at the destination enriches the understanding of rate problems. Mastering such word problems is key in improving problem-solving skills involving speed, time, and distance.