Problem 44 - Prealgebra and Introductory Algebra

Question

On a recent trip, a trucker traveled \(330 \mathrm{mi}\) at a constant rate. Because of road conditions, the trucker then reduced the speed by 25 mph. An additional 30 mi was traveled at the reduced rate. The entire trip took 7 h. Find the rate of the trucker for the first \(330 \mathrm{mi}\)

Step-by-Step Solution

Answer

The trucker's rate for the first 330 miles was 65 mph.

Step 1: Defining the Variables and Writing Down the Known Parameters

Let's denote the trucker's rate for the first 330 miles as \( R \) (in mph). Then the speed during the second part of the trip is \( R-25 \) mph. The time spent in the first part of the trip can be denoted as \( T \) (in hours). The time spent at the second part of the trip is therefore \( 7-T \) (in hours), since the total trip took 7 hours.

Step 2: Writing Equations Based on Known Parameters

Now, we formulate our equations. For the first part of the trip, we use the formula for time and write the equation as \( 330 = R \cdot T \). For the second part of the trip, we use the formula for distance. We know the distance travelled and the speed, so the time spent on this part of the trip will be \( 30 = (R - 25) \cdot (7 - T) \).

Step 3: Solving These Equations

In order to solve these equations, we can choose the substitution or elimination methods. But here substitution is easier. First, derive \( T \) from the first equation as \( T = 330 / R \), and substitute \( T \) into the second equation.

Step 4: Solving for R

After simplifying the resulting equation, solve for \( R \). We get a quadratic equation \( 30R - 205 = 330 \). Solving this equation would give \( R = 65 \) mph

Step 5: Checking the Solution

To assure the solution is correct, substitute the result \( R = 65 \) into the initial equations to make sure they are true. After substituting, check if the total time is indeed equal to 7 hours.

Key Concepts

Constant Rate Travel

In constant rate travel, a traveler moves at a consistent speed without any acceleration or deceleration over a period of time. This notion simplifies calculations and helps us understand how distance and time interact. Constant rate travel is characterized by the relation:

• Distance = Rate × Time

For example, in the trucker's situation, the first 330 miles were traveled at a steady speed of \( R \) mph. The given problem incorporates a change in speed due to road conditions, which is realistically common. However, the initial part of the journey exemplifies constant rate travel, where the trucker maintained consistent speed throughout the 330 miles. Achieving an optimal understanding of this concept is essential, as it sets the foundation for solving more complex travel-related word problems.

Solving Quadratic Equations

Solving quadratic equations is often a necessary step in tackling word problems that involve changes in rates or other variables. These equations typically take the form:\[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants. In the trucker’s problem, solving for the rate \( R \) leads to a quadratic equation that needs to be simplified and solved. Using methods like factoring, completing the square, or applying the quadratic formula helps us to find meaningful values for unknown variables. In this situation, once we set up the equations based on the trucker's journey, we simplify to obtain:

• Original equation: \( 30R - 205 = 330 \)
• Simplified to find \( R \)

This teaches us how changes in travel conditions necessitate employing quadratic equations to find specific speeds or rates.

Distance and Time Relationships

The connection between distance, time, and rate is fundamental to solving travel word problems. By understanding this relationship, we can effectively model situations and analyze different scenarios. In general, the formula \( \, \text{Distance} = \text{Rate} \times \text{Time} \) governs these problems. Breaking it down:

• Distance: The total path traveled, measured in units like miles or kilometers.
• Rate: Speed of travel, usually expressed in miles per hour (mph) or kilometers per hour (kph).
• Time: Duration spent traveling, often measured in hours or minutes.

For the trucker, the journey's total time was 7 hours. The initial equation \( 330 = R \cdot T \) signifies the understanding that different variables are intertwined, while "total time = 7 hours" ensures constraints are followed when solving. Knowing how to manipulate and interpret these relationships is crucial for solving complex travel problems.