8. Each block in a child’s set of building blocks is 15 cm long, 10 cm wide, and 5 cm high. Suppose you put the blocks in a box that is 50 cm long, 35 cm wide, and 30 cm high.
Suppose you arrange the blocks neatly in layers. How many different ways can you layer the blocks? How many blocks fit in the box each way? Which is the best way to pack the blocks? Why?
Can you give a full explination. I don't understant this
The box is 2*15 by 5*10 by 7*5
so, one way would be layers
each 2 blocks long and 5 blocks wide, and 7 layers high
Try factoring the dimensions in other ways, making sure that each block size is one of the factors.
If you must have corresponding dimensions (not rotating the blocks to fit in other ways), then this is the only way to do it, since there are no other multiples that all fit in the same orientation.
how did you get 2,5,7??
To determine the number of ways you can layer the blocks, you need to consider the dimensions of the blocks and the dimensions of the box.
First, let's find out how many blocks can fit in each dimension of the box:
- Length: Divide the length of the box (50 cm) by the length of each block (15 cm):
Number of blocks in length = 50 cm / 15 cm = 3.33 (approx.)
Since you cannot have a fraction of a block, you can fit a maximum of 3 blocks in the length dimension.
- Width: Divide the width of the box (35 cm) by the width of each block (10 cm):
Number of blocks in width = 35 cm / 10 cm = 3.5 (approx.)
Similarly, you can fit a maximum of 3 blocks in the width dimension.
- Height: Divide the height of the box (30 cm) by the height of each block (5 cm):
Number of blocks in height = 30 cm / 5 cm = 6
You can fit a maximum of 6 blocks in the height dimension.
Now, to find the total number of different ways you can layer the blocks, you need to multiply the number of blocks in each dimension:
- Total number of ways = Number of blocks in length x Number of blocks in width x Number of blocks in height
= 3 x 3 x 6
= 54
So, there are 54 different ways you can layer the blocks in the given box.
Next, let's determine how many blocks fit in the box in each way. Since you can fit 3 blocks in the length and width dimensions, and 6 blocks in the height dimension, the number of blocks that fit in the box each way is:
- Number of blocks that fit = Number of blocks in length x Number of blocks in width x Number of blocks in height
= 3 x 3 x 6
= 54
So, in each of the 54 different ways, you can fit a total of 54 blocks in the box.
Now, let's discuss the best way to pack the blocks. Since the total number of blocks (54) is the same in all ways, the best way to pack the blocks would be the most efficient use of space within the box.
In this case, the best way to pack the blocks would be to arrange them neatly in layers, fitting as many blocks as possible in each layer while still maintaining stability. You can start by placing 3 blocks in the length dimension, 3 blocks in the width dimension, and 6 blocks in the height dimension. Then, you can continue to stack additional layers on top until all 54 blocks are packed.
By utilizing the maximum available space in each dimension, this method ensures the most efficient packing of the blocks, minimizing wasted space within the box.
Overall, the best way to pack the blocks is to arrange them neatly in layers, using the maximum available space in each dimension, resulting in a total of 54 blocks packed in the box.
To calculate the number of different ways you can layer the blocks, you need to consider the dimensions of both the blocks and the box.
Let's start by finding the number of blocks that can fit in each dimension. Divide the length of the box (50 cm) by the length of each block (15 cm):
50 cm / 15 cm = 3.33 (approximately)
So, in the length dimension, you can fit a maximum of 3 blocks.
Next, divide the width of the box (35 cm) by the width of each block (10 cm):
35 cm / 10 cm = 3.5 (approximately)
In the width dimension, you can fit a maximum of 3 blocks as well.
Finally, divide the height of the box (30 cm) by the height of each block (5 cm):
30 cm / 5 cm = 6
In the height dimension, you can fit a maximum of 6 blocks.
To find the total number of ways to layer the blocks, you multiply the maximum number of blocks in each dimension:
3 (length) × 3 (width) × 6 (height) = 54
So, there are 54 different ways to arrange the blocks in layers within the given box.
To determine how many blocks fit in the box for each arrangement, you can calculate the volume of the box and the volume of one block. Then, divide the volume of the box by the volume of one block.
The volume of the box is calculated by multiplying its length, width, and height:
50 cm × 35 cm × 30 cm = 52,500 cm³
The volume of one block is calculated by multiplying its length, width, and height:
15 cm × 10 cm × 5 cm = 750 cm³
Finally, dividing the volume of the box by the volume of one block gives us the number of blocks that fit in the box for each arrangement:
52,500 cm³ / 750 cm³ = 70
So, each way of layering the blocks allows you to fit 70 blocks in the box.
Now, let's discuss which is the best way to pack the blocks. Since we calculated that there are 54 different ways to layer the blocks, you need to consider other factors to determine the best way.
Some factors to consider could be stability, ease of access, and efficient use of space. For example, if you want to ensure stability, you may choose to distribute the weight of the blocks evenly across the layers. Additionally, leaving some space for easy access to the blocks might be important.
Ultimately, the best way to pack the blocks depends on your specific requirements and preferences.