Problem 35 - Elementary Algebra

Question

For exercises \(35-36, T=\frac{336 \mathrm{gm}}{R}\) represents the relationship of tire diameter, \(T\); gear ratio, \(g\); speed, \(m\); and revolutions of the tire per minute, \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ g \text { and } m \text { are constant; the relationship of } T \text { and } R $$

Step-by-Step Solution

Answer

The relationship of \(T\) and \(R\) is an inverse variation.

Step 1: Identify Constants and Variables

In the given problem, the equation is provided as: \[ T = \frac{336 \text{ gm}}{R} \] Here, \(g\) and \(m\) are constants, while \(T\) (tire diameter) and \(R\) (revolutions of the tire per minute) are variables.

Step 2: Analyze the Proportional Relationship

Observe the form of the equation: \[ T = \frac{336 \text{ gm}}{R} \] This form suggests that as \(T\) increases, \(R\) must decrease correspondingly, and vice versa, in order to keep the fraction constant.

Step 3: Define Direct and Inverse Variation

Direct variation implies that one variable increases or decreases proportionally with another. Inverse variation implies that one variable increases as the other decreases, following the form \( y = \frac{k}{x} \), where \(k\) is a constant.

Step 4: Compare with Inverse Variation Form

The given equation \( T = \frac{336 \text{ gm}}{R} \) matches the form of inverse variation \( y = \frac{k}{x} \), where \(k = 336 \text{ gm}\). Hence, \(T\) and \(R\) exhibit an inverse relationship.

Key Concepts

Tire Diameter

Tire diameter is a crucial measurement in many automotive calculations. It refers to the full span across the tire, measured from one edge to the other through the center.
This measurement affects various factors including speed, gear ratios, and the number of tire revolutions per minute.
For instance, a larger tire diameter will cover more distance with each revolution, resulting in fewer revolutions needed to maintain the same speed.
Conversely, a smaller tire diameter will require more revolutions to cover the same distance.

• Understanding tire diameter is essential for making accurate mechanical and performance assessments.
• It also helps in modifying or upgrading vehicles for better performance.
• The tire diameter is inversely related to the tire's revolutions per minute when other factors are constant.

Revolutions per Minute

Revolutions per minute (RPM) measures how many times a tire completes a full rotation in one minute. This metric is critical for determining vehicle speed and efficiency.
In the provided equation, RPM is represented as 'R'.
If a tire completes many revolutions per minute, it typically means the vehicle is traveling at a higher speed.
However, the relationship between RPM and tire diameter is inverse. When the diameter increases, the RPM must decrease to keep the speed constant.

• This inverse relationship allows for the adjustment of speed without changing other mechanical components.
• Monitoring RPM can also help in diagnosing mechanical issues or making necessary adjustments for optimal performance.

Proportional Relationship

Proportional relationships describe how one quantity changes in relation to another. Direct variation occurs when both quantities increase or decrease together.
In contrast, inverse variation is when one quantity increases as the other decreases.
In the provided problem, the relationship between tire diameter (T) and revolutions per minute (R) is an example of inverse variation.
The equation \( T = \frac{336 \text{ gm}}{R} \) shows that as T increases, R decreases to keep the product constant.

• Mathematically, this is represented as \( y = \frac{k}{x} \), where k is a constant.
• This principle can be applied in various fields to understand and manipulate relationships between different quantities.
• Recognizing these variations can help in designing better systems and troubleshooting issues effectively.