Question

A train is traveling for 5 hours at a constant rate of \(x\) miles per hour and then travels an additional \(\frac{5}{13}\) hours at a speed of \(\frac{x}{2}\) miles per hour. If the train travels a total of 300 miles during these two segments, which equation could be used to solve for \(x ?\) (A) \(x^{2}+100 x-6,000=0\) (B) \(x^{2}+100 x-300=0\) (C) \(x^{2}+5 x-300=0\) (D) \(3 x^{2}+150 x-6,000=0\)

Short answer

The equation that could be used is \(3x^{2}+150x-6,000=0\).

Step-by-step solution

Calculate Distance for First Segment

The train travels for 5 hours at a speed of \(x\) miles per hour. To find the distance covered during this time, use the formula: \(\text{Distance} = \text{Rate} \times \text{Time}\). Thus, the distance for the first segment is \(5x\).

Calculate Distance for Second Segment

The train travels an additional \(\frac{5}{13}\) hours at a speed of \(\frac{x}{2}\) miles per hour. Again, use the formula \(\text{Distance} = \text{Rate} \times \text{Time}\). Therefore, the distance for the second segment is \(\left(\frac{5}{13}\right) \times \left(\frac{x}{2}\right)\), which simplifies to \(\frac{5x}{26}\).

Set Up the Total Distance Equation

The total distance traveled for both segments is 300 miles. Combine the distances from the two segments to set up the equation: \(5x + \frac{5x}{26} = 300\).

Clear the Fraction

To eliminate the fraction, multiply every term by 26: \(26 \times 5x + 26 \times \frac{5x}{26} = 26 \times 300\), which simplifies to \(130x + 5x = 7,800\).

Simplify the Equation

Combine like terms on the left side of the equation: \(135x = 7,800\).

Finalize the Equation

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Correct Alignment

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Rationalize Necessary Framework Solution Calibration

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Key concepts

Quadratic Equations

Quadratic equations play a vital role in a wide range of mathematical applications, including solving problems that involve curves and parabolas. A quadratic equation is typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our train problem, we're trying to find the speed \(x\) using such an equation. The steps involve setting up an equation based on known quantities and trying to satisfy the equation to make the left-hand side equal to the right-hand side. Solving a quadratic equation usually involves factoring, completing the square, or using the quadratic formula. In these problems, each term balances across the equation with respect to distance, time, and speed. Quadratic equations can provide multiple solutions, which are typically tested to see which fits the context of the problem.

Distance, Rate, and Time Problems

Distance, Rate, and Time problems are a classic type of problem that tests understanding of the relationship between these three variables. The basic formula used here is \(\text{Distance} = \text{Rate} \times \text{Time}\). Understanding this relationship is crucial as it forms the basis for solving such exercises. In our train scenario, the train travels with different speeds during different time intervals, making it necessary to apply this formula separately to each segment. For the first part of the journey, the train covers a total distance of \(5x\) miles in 5 hours, and for the second, \(\left(\frac{5}{13}\right) \times \left(\frac{x}{2}\right)\) miles in \(\frac{5}{13}\) hours. Combining these to equal 300 miles helps us set up the quadratic equation required to solve for \(x\). Thus, understanding the formula \(\text{Distance} = \text{Rate} \times \text{Time}\) and knowing how to break down problems into segments are key solving strategies in these exercises.

Problem-Solving Strategies

Effective problem-solving strategies in mathematics involve breaking down the problem into more manageable parts and using logical reasoning to solve each part. Here, we divided the journey into two time segments, calculated the distance for each, and added these to set up an equation. This systematic approach helps prevent errors and ensures all aspects of the problem are considered.

• Always start by identifying what you know and what you need to find out.
• Use any given formulas to relate the known and unknowns.
• Check each step for logical consistency and mathematical accuracy.
• Consider all possible solutions and test each to see which fits the problem context.

Such strategies not only help solve specific problems but also improve general problem-solving capabilities across diverse mathematical fields.