Another band has 48 speakers. They stack them at least 4 high, but no taller than 6 high. What are all the different arrays they can make?
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To determine all the different arrays the band can make with the 48 speakers, we need to consider the different heights at which they can stack the speakers.
Given that the speakers can be stacked at least 4 high but no taller than 6 high, we can find all the possible arrays by considering each possible height and determining the number of speakers in each height configuration. Let's analyze each possible height individually:
1. If the band stacks the speakers 4 high, we can calculate the total number of arrays as follows:
- The first layer would have 48 speakers.
- For the second layer, there would be 48 - 4 = 44 available speakers.
- Similarly, for the third layer, there would be 44 - 4 = 40 available speakers.
- Finally, for the fourth layer, there would be 40 - 4 = 36 available speakers.
So, in this case, the band can form one array of 4 layers, each containing 48, 44, 40, and 36 speakers.
2. If the band stacks the speakers 5 high, we can calculate the total number of arrays as follows:
- The first layer would have 48 speakers.
- For the second layer, there would be 48 - 5 = 43 available speakers.
- Similarly, for the third layer, there would be 43 - 5 = 38 available speakers.
- For the fourth layer, there would be 38 - 5 = 33 available speakers.
- And for the fifth layer, there would be 33 - 5 = 28 available speakers.
Consequently, in this case, the band can form one array of 5 layers, each containing 48, 43, 38, 33, and 28 speakers.
3. If the band stacks the speakers 6 high, we can calculate the total number of arrays as follows:
- The first layer would have 48 speakers.
- For the second layer, there would be 48 - 6 = 42 available speakers.
- Similarly, for the third layer, there would be 42 - 6 = 36 available speakers.
- For the fourth layer, there would be 36 - 6 = 30 available speakers.
- And for the fifth layer, there would be 30 - 6 = 24 available speakers.
- Finally, for the sixth layer, there would be 24 - 6 = 18 available speakers.
Thus, in this case, the band can form one array of 6 layers, each containing 48, 42, 36, 30, 24, and 18 speakers.
Combining all the possible heights, we find that the band can create a total of three different arrays:
- One array with 4 layers (48, 44, 40, 36 speakers)
- One array with 5 layers (48, 43, 38, 33, 28 speakers)
- One array with 6 layers (48, 42, 36, 30, 24, 18 speakers)