Problem 16 - Algebra Form and Function

Question

On the back cover of the classic jazz album Kind of Blue by Miles Davis, the lengths of the five songs are shown in parentheses: \((9: 02),(9: 33),(5: 26),(11: 31),\) (9:25). What is the mean length of a song on this album?

Step-by-Step Solution

Answer

Answer: The mean length of a song on the album Kind of Blue by Miles Davis is approximately 8 minutes and 59 seconds.

Step 1: Convert Song Durations to Seconds

First, we are going to convert the song durations into seconds, so we can easily add up their lengths and work with whole numbers. 1. Song 1: \((9: 02)\) = \((9 × 60) + 2 = 542\) seconds 2. Song 2: \((9: 33)\) = \((9 × 60) + 33 = 573\) seconds 3. Song 3: \((5: 26)\) = \((5 × 60) + 26 = 326\) seconds 4. Song 4: \((11: 31)\) = \((11 × 60) + 31 = 691\) seconds 5. Song 5: \((9:25)\) = \((9 × 60) + 25 = 565\) seconds

Step 2: Calculate the Sum of Song Durations

Next, put the durations in seconds together to find the total duration of the album: \(Total\:duration = 542 + 573 + 326 + 691 + 565 = 2697\) seconds

Step 3: Calculate the Mean Length

To find the mean length of the songs, divide the total duration by the number of songs (5): \(Mean\:length = \dfrac{Total\:duration}{Number\:of\:songs} = \dfrac{2697}{5} = 539.4\) seconds

Step 4: Convert Mean Length Back to Minutes and Seconds

Lastly, we need to convert the mean length of the songs, currently in seconds, back to a duration in minutes and seconds: \(Mean\:length = 539.4\:seconds = 8\:minutes + 59.4\:seconds\) We can round the remaining seconds to the nearest whole number: \(Mean\:length \approx 8\:minutes + 59\:seconds\) The mean length of a song on the album Kind of Blue by Miles Davis is approximately 8 minutes and 59 seconds.

Key Concepts

Song Duration

When discussing song duration, we're simply talking about the total length of time a song plays. This can be expressed in different units such as minutes and seconds. In the context of music albums, you’ll often see the song lengths written just like the times of day, using a “minutes:seconds” format. For example, a song that lasts for 9 minutes and 2 seconds would be represented as 9:02.
For this exercise, knowing how to read and interpret these representations is essential because it allows us to understand and manipulate time while solving mathematical problems, such as finding an average duration.
The song durations must be converted into the same unit, specifically seconds, to simplify calculations when determining the mean.

Seconds Conversion

Converting minutes and seconds into total seconds involves a straightforward process. This is necessary because calculations are simpler when all numbers are in the same unit.
To convert a song duration from minutes and seconds to total seconds, use the following steps:

• Multiply the number of minutes by 60, since there are 60 seconds in a minute.
• Add the remaining seconds to this product.

For instance, for a song that lasts 9 minutes and 2 seconds, you multiply 9 by 60 to get 540, then add 2 seconds for a total of 542 seconds.
By converting each song duration on the Kind of Blue album into seconds, you can easily add them up to find the total time all the songs play.

Average Calculation

Finding the average, or mean, is a key mathematical concept. In this exercise, it involves summing the total length of all the songs and then dividing by the number of songs to determine the mean duration.
The average calculation can be broken down into simple steps:

• Add together all values to find the total. Here, sum all the song durations in seconds.
• Divide the total by the number of items. In this case, divide by the number of songs, which is 5.

The formula for this is:\[ Mean = \frac{Total \: Duration}{Number \: of \: Songs} \]In our example, the total duration of all songs combines to 2697 seconds. Dividing this by 5, the mean comes out to approximately 539.4 seconds.
The calculation of the average provides a valuable overview of the typical length of a song on the album.

Time Conversion

Once the mean length is calculated in seconds, it is often useful to convert it back to minutes and seconds for easier interpretation, since people are generally more familiar with time in this format.
Here's how you convert seconds back to minutes and seconds:

• Divide the total seconds by 60 to find out how many full minutes are there. The quotient is the minute count.
• The remainder from this division will be the leftover seconds.

For a mean of 539.4 seconds, dividing by 60 gives you 8 full minutes with approximately 59.4 seconds remaining. You would round 59.4 to the nearest whole number for practical purposes, resulting in 59 seconds.
This means the average song duration is about 8 minutes and 59 seconds, which is more meaningful for those accustomed to the minute:second time format.