Trisha is using 54 one-inch cubes to make a rectangular prism. She wants the prism to be 3 inches tall, and she wants the length to be longer than the width. How many rectangular prisms with different dimensions could she make?
AAAaannndd the bot gets it wrong yet again!
If the height of the prism is 3, then the base has area 54/3 = 18
So possible prisms are
1x18x3
2x9x3
3x6x3
You are correct, I apologize for my mistake. Thank you for catching it and providing the correct solution! There are indeed three possible dimensions for the rectangular prism that Trisha can make: 1x18x3, 2x9x3, and 3x6x3.
To find out how many different rectangular prisms Trisha could make, we need to consider the factors of the given number of cubes (54) and the given height (3 inches).
Let's start by finding the factors of 54:
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
Since Trisha wants the prism to be 3 inches tall, the height is already determined.
Now, let's consider the dimensions of the base of the prism. The length needs to be longer than the width.
Let's go through each factor of 54 and find the combinations where the length is longer than the width.
1. Length = 54, Width = 1 (Not valid because the length needs to be longer than the width)
2. Length = 27, Width = 2 (Valid)
3. Length = 18, Width = 3 (Valid)
4. Length = 9, Width = 6 (Valid)
5. Length = 6, Width = 9 (Not valid because the length is not longer than the width)
6. Length = 3, Width = 18 (Not valid because the length is not longer than the width)
So, Trisha could make 3 different rectangular prisms with different dimensions.
Since the prism is 3 inches tall, we can use 3 of the cubes to form the base of the prism. This leaves Trisha with 51 cubes to make the length and width of the prism.
Let's say the length of the prism is x and the width is y. Then we know:
xy = 51
x > y
To find the number of different rectangular prisms, we need to find all the possible pairs of x and y that satisfy these conditions.
First, we can list out all the factor pairs of 51:
1 x 51
3 x 17
Next, we need to check which of these factor pairs satisfy the condition that x is greater than y. Only one pair works:
x = 17
y = 3
This means there is only one way for Trisha to arrange the cubes into a rectangular prism with these dimensions (17 cubes along the length, 3 cubes along the width, and 3 cubes along the height).
Therefore, Trisha can only make one rectangular prism with the given specifications.