another band has 30 speakers. they stack them at least 4 high, but no taller than 6 high. what are all the different arrays they could make?
another band has 48 speakers. they stack them at 4 high, but no taller than 6 high. what are all the different arrays they can make?
To find all the different arrays the band can make with their 30 speakers, we can use a combination of mathematical reasoning and counting principles. Here's how you can approach it step by step:
Step 1: Determine the range of possible speaker heights
Based on the given information, we know that the speakers can be stacked at least 4 high but no taller than 6 high. So, we have a range of speaker heights from 4 to 6.
Step 2: Determine the possible number of speakers in each array
Since we have a total of 30 speakers, each array must have a combination of speaker heights that adds up to 30. Let's consider each possible number of speakers in an array:
- If we have an array with 4-high speakers, we can fit up to 7 arrays (7 x 4 = 28) within the 30 speaker limit.
- If we have an array with 5-high speakers, we can fit up to 6 arrays (6 x 5 = 30) within the limit.
- If we have an array with 6-high speakers, we can fit up to 5 arrays (5 x 6 = 30) within the limit.
Step 3: Find all the different combinations
Now that we know the possible number of speakers for each array height, we can determine all the different combinations by varying the number of arrays we use.
For 4-high speakers, we have 7 options:
- 7 arrays of 4 speakers each
- 6 arrays of 4 speakers each, and 1 array of 2 speakers
- 5 arrays of 4 speakers each, and 1 array of 4 speakers
- 4 arrays of 4 speakers each, and 2 arrays of 2 speakers
- 3 arrays of 4 speakers each, and 3 arrays of 2 speakers
- 2 arrays of 4 speakers each, and 5 arrays of 2 speakers
- 1 array of 4 speakers, and 7 arrays of 2 speakers
For 5-high speakers, we have 6 options:
- 6 arrays of 5 speakers each
- 5 arrays of 5 speakers each, and 1 array of 5 speakers
- 4 arrays of 5 speakers each, and 2 arrays of 5 speakers
- 3 arrays of 5 speakers each, and 3 arrays of 5 speakers
- 2 arrays of 5 speakers each, and 4 arrays of 5 speakers
- 1 array of 5 speakers, and 5 arrays of 5 speakers
For 6-high speakers, we have 5 options:
- 5 arrays of 6 speakers each
- 4 arrays of 6 speakers each, and 1 array of 6 speakers
- 3 arrays of 6 speakers each, and 2 arrays of 6 speakers
- 2 arrays of 6 speakers each, and 3 arrays of 6 speakers
- 1 array of 6 speakers, and 4 arrays of 6 speakers
So, in total, there are 7 + 6 + 5 = 18 different arrays that the band can make with their 30 speakers.