Question

MODELING WITH MATHEMATICS The number \(y\) of songs that can be stored on an MP3 player varies inversely with the average size \(x\) of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB). (See Example 4.) a. Make a table showing the number of songs that will fit on the MP3 player when the average size of a song is \(2 \mathrm{MB}, 2.5 \mathrm{MB}, 3 \mathrm{MB}\), and \(5 \mathrm{MB}\). b. What happens to the number of songs as the average song size increases?

Short answer

As the size of the songs increase, less number of songs can be stored on the MP3 player.

Step-by-step solution

Identify the constant of variation

First, we find the constant \(k\) using the provided values of \(y\) and \(x\) which are 2500 and 4 respectively. Plugging these into the formula \(k = yx\), it comes out to be \(k = 2500 × 4 = 10000\).

Calculate number of songs for different song sizes

Now, we use the \(k\) value we found and the inverse variation formula \(y = k/x\) to find out the number of songs that the MP3 player can hold for different song sizes. That is \(y = 10000/x\). For \(x = 2, 2.5, 3, 5\), we get \(y = 5000, 4000, 3333.33, 2000\) respectively.

Understand the relationship between the numbers of songs and song size

As it can be seen, as the song size increases, the number of songs that can be stored in the MP3 player decreases. This is because, the total storage capacity of the MP3 player is fixed and thus with the increase in the size of the songs, less number of songs can be stored.

Key concepts

Mathematical Modeling

Mathematical modeling is a powerful tool used to describe and analyze real-world situations mathematically. It involves creating equations or functions that represent how different quantities are related. In our exercise about the MP3 player, we are dealing with an inverse variation model.

This means the number of songs (\(y\)) the device can hold varies inversely with the average size of a song (\(x\) MB), indicative of a relationship where as one quantity increases, the other decreases. To build this model, we use the inverse variation equation, which in this case is expressed as \(y = \frac{k}{x}\), where \(k\) is a constant highlighting the total storage capacity when multiplied by the average song size.

Constant of Variation

The constant of variation (\(k\)) in inverse variation plays a crucial role as it remains unchanged regardless of the values of other variables. It is what links two variables inversely; in our example, it relates the number of songs to their sizes. To deduce this constant, we use the fact that an MP3 player can store 2500 songs when the song size is 4MB, giving us the equation \(2500 = k/4\). Solving for \(k\), we find that the constant of variation is 10000. This number represents a fixed capacity of storage in terms of song quantity times song size. Once known, we can compute the possible number of songs (\(y\)) for any given song size (\(x\)) using the formula \(y = \frac{k}{x}\).

Solving Algebraic Equations

Solving algebraic equations is a staple of algebra that involves finding the values of unknowns that make an equation true. Inverse variation problems typically result in one-variable algebraic equations that are straightforward to solve. With our constant of variation (\(k\)) and the inverse variation formula, we just plug in different values for \(x\) to get \(y\). For example, when \(x = 2.5\), we solve for \(y\) in \(y = \frac{10000}{2.5}\) to find that 4000 songs can fit on the MP3 player. Through these simple calculations, algebra brings clarity to how quantities depend on each other in real-life scenarios like storing data on electronic devices.

Understanding Relationships in Algebra

In algebra, it's important to understand how variables relate to each other within an equation or function. Inverse relationships are particularly interesting because they showcase a distinct characteristic: as one variable increases, the other decreases. This inverse variation is seen in the MP3 player example, where the relationship between the song size and the number of songs clearly displays this pattern. As the size of the songs (\(x\)) increases, our model shows that the number of songs (\(y\) that the MP3 player can store decreases proportionally. This understanding helps us predict outcomes and make decisions based on how altering one variable will affect another.