one band has 30 speakers. they stack them at least 3 high, but no taller than 8 high. what are all the different arrays they can make?
9p+q+6p+5q+5p-5q
To find all the different arrays that can be made with the speakers, we need to consider the number of speakers that can be stacked at different heights between 3 and 8.
We'll start by listing out the possible heights for stacking speakers: 3, 4, 5, 6, 7, 8.
For each height, we'll determine the number of speakers that can be stacked and check if any arrangements are possible. Here's how we can calculate it:
1. Height 3:
- The maximum number of speakers that can be stacked at a height of 3 is 10 (30 speakers divided by 3).
- We need at least 3 speakers to form an array, so we can have arrays of 3, 4, 5, 6, 7, 8, 9, or 10 speakers.
2. Height 4:
- The maximum number of speakers that can be stacked at a height of 4 is 7 (30 speakers divided by 4).
- We need at least 3 speakers to form an array, so we can have arrays of 3, 4, 5, 6, or 7 speakers.
3. Height 5:
- The maximum number of speakers that can be stacked at a height of 5 is 6 (30 speakers divided by 5).
- We need at least 3 speakers to form an array, so we can have arrays of 3, 4, 5, or 6 speakers.
4. Height 6:
- The maximum number of speakers that can be stacked at a height of 6 is 5 (30 speakers divided by 6).
- We need at least 3 speakers to form an array, so we can have arrays of 3, 4, or 5 speakers.
5. Height 7:
- The maximum number of speakers that can be stacked at a height of 7 is 4 (30 speakers divided by 7).
- We need at least 3 speakers to form an array, so we can have arrays of 3 or 4 speakers.
6. Height 8:
- The maximum number of speakers that can be stacked at a height of 8 is 3 (30 speakers divided by 8).
- We need at least 3 speakers to form an array, so we can have arrays of 3 speakers.
So, considering all possible heights, the different arrays that can be made are:
- Arrays of 3, 4, 5, 6, 7, 8, 9, 10 speakers at a height of 3.
- Arrays of 3, 4, 5, 6, or 7 speakers at a height of 4.
- Arrays of 3, 4, 5, or 6 speakers at a height of 5.
- Arrays of 3, 4, or 5 speakers at a height of 6.
- Arrays of 3 or 4 speakers at a height of 7.
- Arrays of 3 speakers at a height of 8.
Therefore, there are multiple possible arrays that can be made with different heights.