Problem 24 - Algebra 2 and Trigonometry

Question

In \(23-28,\) write an equation of the direct variation described. The distance in miles, \(d\) , that Mr. Spencer travels is directly proportional to the length of time in hours, \(t,\) that he travels at 35 miles per hour.

Step-by-Step Solution

Answer

The equation is \(d = 35t\).

Step 1: Understand Direct Variation

Direct variation implies that one variable is a direct constant multiple of another variable. In this case, the distance \(d\) varies directly with time \(t\). The equation for direct variation can be written as \(d = k \cdot t\), where \(k\) is the constant of proportionality.

Step 2: Identify the Constant of Proportionality

Since Mr. Spencer travels at a constant speed of 35 miles per hour, this speed is indeed the constant of proportionality \(k\). Thus, \(k = 35\).

Step 3: Write the Equation of Direct Variation

Substitute the constant of proportionality into the direct variation formula to get the specific relationship: \(d = 35 \cdot t\).

Key Concepts

Equation of Direct Variation

The equation of direct variation is a simple way to express how two variables are directly related. In this context, direct variation means that the distance Mr. Spencer travels, known as \(d\), is directly dependent on the time \(t\) he spends traveling. This relationship is captured by the equation:

• \( d = k \cdot t \)

Here:

• \(d\) represents the distance traveled,
• \(t\) represents the time taken,
• \(k\) is the constant that multiplies \(t\) to get \(d\).

This equation implies that as one variable (time) changes, the other (distance) changes in a way that maintains a consistent ratio. That ratio is the constant \(k\).

Constant of Proportionality

The constant of proportionality, denoted as \(k\), is a crucial factor in equations of direct variation. It signifies the consistent rate at which the two variables change relative to each other. In our example, Mr. Spencer is traveling at a speed of 35 miles per hour. This speed is the constant of proportionality:

• \( k = 35 \)

This means, for every hour \(t\) that Mr. Spencer drives, he covers 35 miles. Hence, the value of \(k\) directly influences the relationship between distance and time. Calculating distance becomes straightforward by multiplying the time \(t\) by this constant rate. This simplicity and reliability make the constant of proportionality a fundamental concept in equations involving direct relationships like speed and travel time.

Distance-Time Relationship

The distance-time relationship in this scenario clearly explains how far Mr. Spencer travels depending on the length of time he drives. Given the direct variation equation \(d = 35 \cdot t\), you see that the distance \(d\) increases as the time \(t\) increases, showcasing a perfect linear relationship.Some key points about the distance-time relationship include:

• As the time \(t\) doubles, so does the distance \(d\).
• If no time is spent traveling (\(t = 0\)), no distance is covered (\(d = 0\)).
• Every additional hour of travel covers an extra 35 miles, thanks to the constant of proportionality (35 miles per hour).

This relationship can be easily visualized on a graph, where the x-axis might represent time and the y-axis represents distance. The graph would show a straight line starting at the origin (0,0) with a slope of 35, demonstrating the constant speed of travel.