Problem 21 - College Algebra

Question

Fiora starts riding her bike at \(20 \mathrm{mi} / \mathrm{h}\). After a while, she slows down to \(12 \mathrm{mi} / \mathrm{h}\), and maintains that speed for the rest of the trip. The whole trip of \(70 \mathrm{mi}\) takes her \(4.5 \mathrm{~h}\). For what distance did she travel at \(20 \mathrm{mi} / \mathrm{h}\) ?

Step-by-Step Solution

Answer

Fiora traveled 40 miles at 20 mi/h.

Step 1: Define the Variables

Let \( x \) be the distance Fiora traveled at \( 20 \text{ mi/h} \). Consequently, the distance she traveled at \( 12 \text{ mi/h} \) would be \( 70 - x \) since the total distance is 70 miles.

Step 2: Create the Time Equation

The time taken to travel the distance at \( 20 \text{ mi/h} \) is \( \dfrac{x}{20} \) hours, and the time taken at \( 12 \text{ mi/h} \) is \( \dfrac{70 - x}{12} \) hours. Since the total time is 4.5 hours, set up the equation: \[\frac{x}{20} + \frac{70 - x}{12} = 4.5\]

Step 3: Solve for x

To solve the equation, first find a common denominator for the fractions, which is 60. Rewrite the equation:\[\frac{3x}{60} + \frac{5(70 - x)}{60} = 4.5\]Multiply both sides by 60 to clear the fractions:\[3x + 5(70 - x) = 270\]Expand and simplify the equation:\[3x + 350 - 5x = 270\]Combine like terms:\[-2x + 350 = 270\]Solve for \( x \):\[-2x = 270 - 350\]\[-2x = -80\]\[x = 40\]

Step 4: Interpret the Solution

Since \( x = 40 \), Fiora traveled 40 miles at a speed of \( 20 \text{ mi/h} \).

Key Concepts

Algebraic Equations

In distance-time problems, algebraic equations are essential tools that help us organize and solve for unknowns. In this specific exercise, we defined a variable to represent the unknown distance Fiora traveled at her initial speed of 20 miles per hour. By introducing \( x \) for this distance, we established a concrete starting point for developing our solution. Creating equations involves:

• Identifying what needs to be solved (the distance \( x \) at 20 mi/h).
• Expressing other known quantities in terms of the variable (e.g., the remainder of the trip is \( 70 - x \)).

Algebraic equations in these problems not only allow us to express relationships between quantities, such as distance and time but also help to systematically break down the problem into manageable steps. In our problem, once the initial equation is framed, it becomes a matter of arithmetic and algebra to find the solution.

Speed and Velocity

Speed is a crucial aspect of understanding distance-time problems. It's the rate at which an object covers distance, and we often express it as distance per time, like miles per hour (mi/h) in this issue. Unlike velocity, which includes direction, speed only considers how fast an object is moving regardless of direction.Fiora's journey can be analyzed in terms of speed in the following ways:

• Initial speed: 20 mi/h — Fast pace to cover the initial part of her journey.
• Reduced speed: 12 mi/h — Slower pace that she maintains until the end.

By analyzing speeds, students understand that different distances are covered over time intervals depending on speed. The key here is to realize that faster speeds result in shorter travel times for the same distance, which allowed us to set up our equation connecting distance and time through speed: \( \, \text{time} = \frac{\text{distance}}{\text{speed}} \). This relationship simplifies the problem-solving process, making calculations more intuitive.

Problem-Solving Steps

The process of tackling a distance-time problem can be methodically approached by following guided steps, which ensure you handle the problem's intricacies without getting overwhelmed.The steps include:

• **Defining Variables:** Clearly mark what each symbol represents. Let \( x \) be the initial distance traveled at the faster speed.
• **Setting Up Equations:** Convert word problems into mathematical equations. Relate distance traveled at different speeds with the time taken using equations.
• **Solving Equations:** Manipulate the equation to isolate the variable, using steps like finding a common denominator and clearing fractions.
• **Interpreting Results:** Convert algebraic solutions back into the context of the question. Ensure the solution makes sense physically — e.g., does 40 miles at 20 mi/h fit the overall travel?
  

Good problem-solving isn't only about math. It's about breaking a problem down, keeping track of units, and ensuring each step logically follows from the last. This structured approach enhances problem-solving skills and builds confidence in dealing with more complex scenarios.