Problem 36 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 356 to solve. Do you still own records, or are you strictly a CD person? Record owners claim that the quality of sound on good vinyl surpasses that of a CD, although this is up for debate. This, however, is not debatable: The number of revolutions a record makes as it is being played is directly proportional to the time that it is on the turntable. A record that lasted 3 minutes made 135 revolutions. If a record takes 2.4 minutes to play, how many revolutions does it make?

Step-by-Step Solution

Answer

A record that plays for 2.4 minutes makes 108 revolutions.

Step 1: Understand the Problem and Identify Key Information

First, note down the key information from the problem: a record that lasted 3 minutes made 135 revolutions. This means each minute corresponds to 45 revolutions since \(135 ÷ 3 = 45\). We also know that the time for another record is 2.4 minutes and we need to find out how many revolutions it makes.

Step 2: Set Up the Direct Variation Equation

Direct variation problems can be solved using the equation \(y = kx\), where \(y\) and \(x\) are the two variables that vary directly and \(k\) is the constant of variation. In this situation, the number of revolutions \(y\) varies directly with time \(x\), and \(k\) will be the number of revolutions per minute (45 revolutions per minute). So our equation looks like this: \(y = 45x\).

Step 3: Plug In the Known Value and Solve

Now we can insert 2.4 minutes into our equation to calculate the number of revolutions: \(y = 45 * 2.4\). It results in \(y = 108\). This means a record that plays for 2.4 minutes makes 108 revolutions.

Key Concepts

Variation Problems

When we talk about variation problems, we're exploring how one quantity changes in relation to another. There are usually two types of variation—direct and inverse. In direct variation, as one variable goes up, so does the other. Conversely, in inverse variation, as one increases, the other decreases.

In the problem we're dealing with, we're focusing on direct variation. This means that the number of revolutions a record makes is directly proportional to the time it spends playing. To solve these types of problems, we can use the equation \( y = kx \), where \( y \) represents one variable (in our case, revolutions), \( x \) represents the other variable (time), and \( k \) is a constant that relates the two.


**Steps to Solve a Variation Problem:**

• Identify the type of variation—direct or inverse.
• Create an equation that relates the variables.
• Use given values to find the constant of variation.
• Substitute the constant back into the equation to solve for unknowns.

Understanding these basics will make other variation problems a breeze to tackle!

Constant of Proportionality

The constant of proportionality, often denoted as \( k \), is a crucial element in direct variation problems. It's what makes the relationship between two variables consistent. Think of it as a "rate" that doesn't change no matter how the variables fluctuate.

In the example from the problem, we calculate the constant of proportionality by looking at the given information: 3 minutes leads to 135 revolutions. Hence, the constant \( k = \frac{135}{3} = 45 \) revolutions per minute. This constant tells us that for every minute the record plays, it will spin 45 times.

The usefulness of \( k \) becomes apparent when you have a situation where one of the variables changes. As long as your situation fits the same pattern (direct variation), you can always calculate unknown values by substituting \( k \) back into the variation equation \( y = kx \).

If you ever feel lost in a direct variation problem, start by identifying \( k \). It's your key to unlocking the solution every time.

Revolutions and Time Relationship

Let's dive deeper into how the number of revolutions is connected to time on a turntable in the form of a direct variation problem. This is a perfect example of how math models real-world scenarios, like listening to records.

In the given problem, we're essentially told that revolutions (\( y \)) and time (\( x \)) have a direct, linear relationship, meaning as one increases, the other does in a consistent manner. We already found that the constant of proportionality (\( k \)) is 45 revolutions per minute. With that, our relationship can be expressed with \( y = 45x \).

For a 2.4-minute playtime, this simple equation tells us exactly how many revolutions are made. Multiply the time by 45 revolutions per minute and you get \( y = 108 \) revolutions.

This relationship shows the power of mathematical modeling in predicting outcomes based on given inputs, and knowing this can save you loads of time when performing similar calculations in other contexts, such as rotations of machinery or even astronomical cycles!