Jada is using 84 one-centimeter cubes to build a rectangular prism. She wants the height of the rectangular prism to be 4 centimeters. She also wants the width to be shorter than the length. How many rectangular prisms with different dimensions could Jada build?
AAAaannndd the bot gets it wrong yet again!
If the volume is 84 cm^3 and the height is 4 cm, then the area of the base must be 84/4 = 21 cm^2
So the base must be either
3x7
or
1x21
To find the number of different rectangular prisms Jada could build, we need to consider the dimensions and use some mathematical principles.
First, let's analyze the dimensions of the rectangular prism. Based on the information given, we know that:
- The height of the prism is fixed at 4 centimeters.
- The length should be greater than the width.
To simplify the problem, let's assign variables to the length, width, and depth (or height) of the prism. Let's say:
- Length = L centimeters
- Width = W centimeters
- Height = 4 centimeters
Since the total number of cubes Jada has is 84, we can now set up an equation based on the number of cubes used to build the prism:
Number of cubes used = Length × Width × Height
Plugging in the values:
84 = L × W × 4
Simplifying the equation:
21 = L × W
Now, let's determine how many pairs of positive integers (L, W) satisfy this equation. Since the length should be greater than the width, we can generate the pairs in the following way:
(1, 21), (3, 7), (7, 3), (21, 1)
We have four valid pairs. It's important to note that the order of length and width matters, so (1, 21) and (21, 1) are considered different pairs.
Therefore, Jada could build four different rectangular prisms with the given conditions.
We know that the height of the rectangular prism is 4 centimeters, so we need to divide the 84 one-centimeter cubes into two dimensions that have a product of 4 (since the height is already determined).
One way to do this is to list out all the possible pairs of dimensions that have a product of 4:
1 x 4
2 x 2
For each pair of dimensions, we need to figure out how many ways we can arrange the 84 cubes to build the rectangular prism. We can use combinations for this:
For the pair of dimensions 1 x 4, we have a total of 5 spaces (4 columns and 1 row, or 4 rows and 1 column) to fill with cubes. We need to choose 3 of these spaces to fill with cubes (since 1 space will be filled by the 4 cubes stacked on top of each other for the height). So the number of combinations is:
5 choose 3, or 5C3, which is equal to 10.
For the pair of dimensions 2 x 2, we have a total of 6 spaces (2 columns and 3 rows, or 2 rows and 3 columns) to fill with cubes. We need to choose 2 of these spaces to fill with cubes (since 2 spaces will be filled by the 4 cubes stacked on top of each other for the height). So the number of combinations is:
6 choose 2, or 6C2, which is equal to 15.
Therefore, Jada can build a total of 10 + 15 = 25 rectangular prisms with different dimensions using the 84 one-centimeter cubes.
I apologize for my mistake in the previous response. You are correct that the possible dimensions of the rectangular prism are either 3x7 or 1x21, since these are the only dimensions that have an area of 21 cm^2 when multiplied by the height of 4 cm.
To find the total number of different rectangular prisms that Jada can build, we need to consider the two possible dimensions and the different ways that the 84 cubes can be arranged within each dimension.
For the dimension 3x7, we have a total of 10 spaces (3 columns and 7 rows, or 3 rows and 7 columns) to fill with cubes. We need to choose 6 of these spaces to fill with cubes (since 4 spaces will be filled by the 4 cubes stacked on top of each other for the height). So the number of combinations is:
10 choose 6, or 10C6, which is equal to 210.
For the dimension 1x21, we have a total of 22 spaces (1 column and 21 rows, or 1 row and 21 columns) to fill with cubes. We need to choose 3 of these spaces to fill with cubes (since 1 space will be filled by the 4 cubes stacked on top of each other for the height). So the number of combinations is:
22 choose 3, or 22C3, which is equal to 1540.
Therefore, Jada can build a total of 210 + 1540 = 1750 rectangular prisms with different dimensions using the 84 one-centimeter cubes.