Problem 36 - 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? $$ g \text { and } R \text { are constant; the relationship of } T \text { and } m \text {. } $$

Step-by-Step Solution

Answer

The relationship is a direct variation.

Step 1: Understand the given formula

The given formula is \[T = \frac{336 \text{ gm}}{R} \]. Here, T is the tire diameter, and R is the revolutions per minute of the tire. We need to understand how T and m relate when g and R are constant.

Step 2: Identify the constants

\(g\) (gear ratio) and \(R\) (revolutions per minute) are given as constants. This means we should treat them as fixed values while analyzing the relationship between \(T\) and \(m\) (speed).

Step 3: Consider the formula with constant terms

When \(g\) and \(R\) are constants, the formula simplifies to \[T = \frac{336 \text{ gm}}{R} \]. Substitute \(R\) with a constant value. This shows \(T\) is directly proportional to speed \(m\).

Step 4: Determine the type of variation

Since \(T\) is proportional to \(336/R\) and \(R\) is constant, this means \(T\) changes directly with \(m\) (i.e., as \(m\) increases, \(T\) increases proportionally). Hence, this is a direct variation.

Key Concepts

Tire Diameter

Tire diameter (T) is the distance across the widest part of a tire. It is an important factor because it affects how far a vehicle travels with each rotation of the tire.

If a tire has a larger diameter, it will cover more ground with each revolution compared to a tire with a smaller diameter. This is because the circumference, which is the distance around the edge of the tire, is larger.

To find the tire diameter using the formula given in the exercise, we simply need to know the revolutions per minute (R) and treat the other factors as constants. The formula shows that if revolutions per minute are constant, the tire diameter will also stay constant if the speed remains unchanged.

Gear Ratio

The gear ratio (g) is the ratio of the rotation speeds of two gears that are interlocked. In vehicles, it is a crucial concept because it determines how effectively the power from the engine is used.

A higher gear ratio means that the engine must turn several times to turn the wheels once. Conversely, a lower gear ratio means the wheels turn more times per engine revolution.

In the given formula, the gear ratio is treated as a constant. This means it does not change regardless of other variables. By keeping the gear ratio constant, we can better examine the direct relationship between tire diameter and speed.

Revolutions per Minute (RPM)

Revolutions per minute (R) refers to how many times a tire completes a full rotation in one minute. It is a helpful measure of how fast a tire is rotating.

The formula shows that revolutions per minute are inversely related to tire diameter when speed and gear ratio are held constant. This means if the number of revolutions per minute increases, the tire diameter must decrease to maintain the same speed.

Understanding revolutions per minute is vital for interpreting how changes in speed or physical characteristics of a tire affect one another. For example, a vehicle going faster will have higher revolutions per minute if the tire diameter remains the same.