Problem 45 - Prealgebra and Introductory Algebra

Question

Commuting from work to home, a lab technician traveled \(10 \mathrm{mi}\) at a constant rate through congested traffic. Upon reaching the expressway, the technician increased the speed by 20 mph. An additional 20 mi was traveled at the increased speed. The total time for the trip was 1 h. At what rate did the technician travel through the congested traffic?

Step-by-Step Solution

Answer

The lab technician traveled through the congested traffic at a speed of 10 mph.

Step 1: Define the variables

Let's denote x as the speed of the lab technician when travelling through the congested traffic. Considering that the speed on the highway was increased by 20mph, hence, the speed on the highway becomes \(x+20\) mph.

Step 2: Establish the equation

The total travel time 1 hour is the sum of the time spent in traffic and the time spent on the expressway. Since time = distance / speed, the time taken in traffic is \(10/x\) hours, and the time taken on the highway is \(20/(x + 20)\) hours. Therefore, the equation to be solved is: \(10/x + 20/(x+20) = 1\)

Step 3: Solve the equation

The equation is a rational equation and can be solved by first multiplying through by \(x(x+20)\), which eliminates the denominators, leading to \(10(x+20)+20x = x(x+20)\) or just \(x^2 + 40x - 400 = 0\). This is a quadratic equation, and it can be solved by factoring to get \((x-10)(x+40) = 0\). This gives two solutions \(x=10, x=-40\). However, speed can't be negative, therefore, the technician travelled through the congested traffic at a speed of 10 mph.

Key Concepts

Quadratic Equations

Quadratic equations are algebraic equations of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. In the exercise, the equation \(x^2 + 40x - 400 = 0\) is a quadratic equation. To solve it, we can use various methods like factoring, the quadratic formula, or completing the square.- **Factoring**: This method involves expressing the quadratic equation as a product of two binomials, such as \((x-10)(x+40) = 0\). From this, we obtain the solutions for \(x\), which are \(x = 10\) and \(x = -40\). As speed cannot be negative, \(x = 10\) is the solution.- **Quadratic Formula**: This is a universal method given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), suitable for all quadratic equations, especially when factoring is difficult.- **Completing the Square**: This technique rearranges the equation to form a perfect square trinomial, simplifying the solution process.Each method has its merits, but factoring can be the quickest when integers are involved. Understanding these techniques is vital in various mathematical problems beyond rate equations.

Distance-Speed-Time Relationships

The relationship between distance, speed, and time plays a crucial role in solving rate problems. The formula is simple yet powerful: \(\text{Distance} = \text{Speed} \times \text{Time}\). From this fundamental relationship, we can derive:- **Time**: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)- **Speed**: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)In the exercise, the technician's travel involved two parts: through congested traffic and on the expressway. By separating the journey into these sections, we used the formula \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\) to determine the time taken for each part.- **Congested Traffic**: Total distance was 10 miles, with speed \(x\). Thus, the time was \(\frac{10}{x}\).- **Expressway**: Distance here was 20 miles, speed \(x + 20\). Hence, time was \(\frac{20}{x + 20}\).The sum of these times equals the total travel time of 1 hour. This breakdown simplifies the problem by allowing us to understand each section's contribution to the journey.

Rational Equations

Rational equations involve variables in the denominator and are solved by finding a common denominator. In the exercise, our primary equation was \(\frac{10}{x} + \frac{20}{x+20} = 1\).Here's a clear approach to solving this type of equation:- **Clear the Denominators**: Multiply through by the common denominator, which is \(x(x+20)\), to eliminate fractions. This simplifies the equation to \(10(x+20) + 20x = x(x+20)\).- **Simplify Further**: After multiplying and combining like terms, we arrive at a solvable quadratic equation: \(x^2 + 40x - 400 = 0\).- **Solve the Quadratic Equation**: With the equation simplified, we can use factoring or other methods to find \(x\).Rational equations can seem challenging, especially with multiple fractions. However, by clearing the denominators and simplifying, they become manageable. Understanding this process is valuable for tackling a range of mathematical problems where rational expressions play a role.