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?
i still dont understand
35 = 5*7 so only the 5cm blocks will work in that dimension
30 = 15*2 = 10*3 so there are various ways to work those numbers
See what you can do.
To determine the number of different ways you can layer the blocks, we need to calculate the number of arrangements possible for each dimension.
For the length, the box is 50 cm long and the block is 15 cm long. Thus, the number of arrangements for the length of the block is 50/15 = 3.
For the width, the box is 35 cm wide and the block is 10 cm wide. Therefore, the number of arrangements for the width of the block is 35/10 = 3.5. However, since the blocks must be arranged neatly, the width arrangement can only be a whole number. Therefore, the number of arrangements for the width is 3.
For the height, the box is 30 cm high and the block is 5 cm high. Thus, the number of arrangements for the height of the block is 30/5 = 6.
To calculate the total number of different ways to layer the blocks, we multiply the number of arrangements for each dimension: 3 (length) x 3 (width) x 6 (height) = 54.
Next, let's determine the number of blocks that fit in the box for each way of layering. To do this, we calculate the volume of the box and the volume of one block.
The volume of the box is given by its length multiplied by its width multiplied by its height: 50 cm x 35 cm x 30 cm = 52,500 cm³.
The volume of one block is given by its length multiplied by its width multiplied by its height: 15 cm x 10 cm x 5 cm = 750 cm³.
To calculate the number of blocks that fit in the box, we divide the volume of the box by the volume of one block: 52,500 cm³ / 750 cm³ = 70.
Therefore, each way of layering the blocks can accommodate 70 blocks.
To find the number of different ways you can layer the blocks, we can use the concept of permutations because the order in which the blocks are layered matters.
First, let's calculate the number of blocks that can fit in each dimension of the box:
- Length: The box is 50 cm long, and each block is 15 cm long. Therefore, the number of blocks that can fit in the length dimension is 50 cm / 15 cm = 3 blocks.
- Width: The box is 35 cm wide, and each block is 10 cm wide. So, the number of blocks that can fit in the width dimension is 35 cm / 10 cm = 3 blocks.
- Height: The box is 30 cm high, and each block is 5 cm high. Hence, the number of blocks that can fit in the height dimension is 30 cm / 5 cm = 6 blocks.
Now, let's determine the number of different ways you can layer the blocks. Since we have 3 layers in the length dimension, 3 layers in the width dimension, and 6 layers in the height dimension, we need to multiply these numbers together:
Number of ways = 3 layers (length) * 3 layers (width) * 6 layers (height) = 54 ways.
Therefore, there are 54 different ways you can layer the blocks.
To figure out the number of blocks that fit in the box each way, simply multiply the number of blocks in each dimension:
Number of blocks = 3 blocks (length) * 3 blocks (width) * 6 blocks (height) = 54 blocks.
So, each way you layer the blocks, you can fit 54 blocks in the box.