Problem 33 - Algebra 1: Concepts and Skills

Question

DRIVING DISTANCE You are driving across the country at an average speed of 65 miles per hour. Using an appropriate formula, find the distance you travel in 4 hours.

Step-by-Step Solution

Answer

The distance travelled is 260 miles.

Step 1: Identification of Given Values

The problem provides two key pieces of information: 1. The average speed of travel, which is 65 miles per hour. 2. The time spent travelling, which is 4 hours.

Step 2: Understand and Apply the Formula

The formula to calculate distance is \( \text{Distance} = \text{Speed} \times \text{Time} \). Here, speed = 65 mph and time = 4 hours.

Step 3: Calculate the Distance

Substitute the given values into the formula: \( \text{Distance} = 65 \times 4 \)

Key Concepts

Average Speed

Understanding average speed is crucial, as it serves as one of the foundational concepts in motion-related problems. Average speed, as the name suggests, is the average velocity of a moving object over a given period. It is the total distance traveled divided by the total time taken to cover that distance. When calculating the average speed, we take the sum of the different speeds at which an object has traveled and divide it by the number of speed measurements.

In algebraic terms, the formula for average speed, \( v_{avg} \), can be expressed as \[ v_{avg} = \frac{Total\text{{ space }}Distance}{Total\text{{ space }}Time} \]. In the exercise provided, we're told the average speed outright: 65 miles per hour (mph), indicating a steady pace. When dealing with exercises involving average speed, it's essential to ensure that the units of distance and time match up; otherwise, additional conversions may be needed.

Time Distance Calculation

Time distance calculations are fundamental in understanding how objects move over a period. Such problems often involve calculating one of three variables: distance, time, or speed, given the other two. The relationship among these three variables is elegantly captured in the simple yet powerful formula, \( Distance = Speed \times Time \).

Applied to real-life situations such as planning trips or understanding travel times, mastering these calculations is incredibly practical. When the speed remains consistent, calculating distance becomes a straightforward multiplication, as shown in the step-by-step solution. If you drive for 4 hours at a constant average speed of 65 mph, you can easily calculate the distance traveled by multiplying these values together, yielding a result that serves both as an exercise in algebra and a practical life skill.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. They are fundamental in translating real-world situations into mathematical form, allowing for calculations and problem-solving. In the context of the driving distance problem, the expression \( 65 \times 4 \) is a simple algebraic expression representing the distance covered when traveling at 65 mph for 4 hours.

To dive further into such expressions, one might encounter variables and constants. In our example, '65' is a constant representing the average speed in mph, while '4' is another constant representing the time in hours. More complex expressions might involve variables representing unknown quantities, which can be solved for given additional information. Students engaging with algebraic expressions learn to operate with unknowns, manipulate terms, and unlock the answers to various problems inspired by real-life scenarios.