Problem 7 - Algebra 2 and Trigonometry

Question

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. A driver travels for 4 hours between stops covering \(d\) miles at a rate of \(r\) miles per hour.

Step-by-Step Solution

Answer

The distance \(d\) and rate \(r\) vary directly.

Step 1: Identify Relationship Type

First, identify whether the variables in this scenario are related by direct or inverse variation. Direct variation implies that if one variable increases, the other also increases proportionally and is expressed as \( y = kx \), where \( k \) is a constant. Inverse variation means that as one variable increases, the other decreases, expressed as \( y = \frac{k}{x} \).

Step 2: Understand the Variables Involved

In this scenario, we have three variables: time \( t = 4 \) hours, distance \( d \) miles, and rate \( r \) miles per hour. The distance \( d \) is calculated using the formula \( d = rt \), which implies direct variation between \( d \) and \( r \) when \( t \) is constant.

Step 3: Determine the Relationship

Since the equation \( d = rt \) defines how distance varies with rate, and the time \( t \) is a constant 4 hours, it shows that distance \( d \) and rate \( r \) vary directly. If \( r \) increases, \( d \) must also increase for the same period, and vice versa.

Key Concepts

Distance-Rate-Time Relationship

The connection between distance, rate, and time is fundamental in understanding motion and travel scenarios. This relationship is encapsulated in the equation \(d = rt\), where \(d\) represents distance, \(r\) denotes rate (or speed), and \(t\) is the time taken. This straightforward formula describes how far something travels over a certain period when moving at a constant speed.

To expand on this concept, consider the following examples:

• If you drive at a speed of 60 miles per hour for 2 hours, the total distance covered can be calculated as: \(d = 60 \times 2 = 120\) miles.
• Alternatively, if you cover 100 miles in 4 hours, your speed or rate can be determined as: \(r = \frac{d}{t} = \frac{100}{4} = 25\) miles per hour.

By understanding this relationship, you can solve for any one of the variables as long as the other two are known. This makes it a crucial formula in many real-life applications, from planning travel routes to understanding vehicle dynamics.

Inverse Variation

Inverse variation describes a situation where one variable increases while the other decreases. This is different from direct variation, where both variables either increase or decrease together. An inverse relationship can be written as \(y = \frac{k}{x}\), where \(k\) is a constant.

For example, if the speed of a vehicle increases, the time taken to travel a fixed distance decreases. In this case, speed and time demonstrate inverse variation. When speed doubles, the time taken is halved, assuming the distance remains constant.

To better understand inverse variation, here are a few points:

• The product of the variables is always the same. If \(x\) increases, \(y\) must decrease such that \(xy = k\). For instance, if \(x = 2\) and \(y = 5\), their product \(k\) is 10. If \(x\) becomes 5, \(y\) needs to be 2 to maintain the product of 10, illustrating the inverse relationship.
• Inverse variation is found in scenarios where rates change over time, like in speed and time calculations.

Recognizing inverse variation helps in identifying how two variables counterbalance each other's effects, which is key in fields like physics and engineering.

Proportional Relationships

Proportional relationships are characterized by a constant ratio between two variables. This means if one variable changes, the other changes at a consistent rate, which is often observed in direct variation scenarios, expressed as \(y = kx\), where \(k\) is the constant of proportionality.

Here's how you can identify and work with proportional relationships:

• If \(y\) is directly proportional to \(x\), then any change in \(x\) results in a proportional change in \(y\). For example, if \(y = 3x\), and \(x\) is 2, \(y\) will be 6. If \(x\) changes to 4, \(y\) becomes 12.
• This relationship implies that the ratio \(\frac{y}{x}\) remains constant, equal to \(k\).
• Proportional relationships simplify complex problems, allowing predictions and calculations to be made with ease, especially in business and economics where such relationships are prevalent.

Understanding proportional relationships equips students to tackle a broad range of mathematical problems, making them an essential part of the math curriculum.