Answer:
Explanation:
To find out how many different rectangular prisms are possible with a volume of 40 cubic inches using only whole number dimensions, we need to consider the factors of 40 and the ways they can be combined to create the dimensions of the rectangular prism.
The factors of 40 are:
1, 2, 4, 5, 8, 10, 20, 40
These factors represent the possible values for the length, width, and height of the rectangular prism. To count the different prisms, we'll consider all the possible combinations of these factors. Keep in mind that for each combination, we can arrange the dimensions in different ways to create distinct prisms.
Let's calculate the possibilities:
1. \(1 \times 1 \times 40\) (and its variations by permuting the dimensions) - 3 possibilities.
2. \(1 \times 2 \times 20\) (and its variations) - 6 possibilities.
3. \(1 \times 4 \times 10\) (and its variations) - 6 possibilities.
4. \(2 \times 2 \times 10\) (and its variations) - 3 possibilities.
5. \(2 \times 5 \times 4\) (and its variations) - 6 possibilities.
Adding up these possibilities: \(3 + 6 + 6 + 3 + 6 = 24\)
So, there are 24 different rectangular prisms possible with a volume of 40 cubic inches using only whole number dimensions.