Question
Multiply the edge lengths of a right rectangular prism with length 12 , width 112 , and height 34 to show that the product is the same as the volume you found by packing the prism with unit cubes with an edge length of 14 inch. Cube method: Number of cubes: 36 Volume of each cube: 14×14×14=164 Volume of the prism: v=36×164 =3664 =916 cubic units (1 point) Responses 12×112×34 Start Fraction 1 over 2 End Fraction times Start Fraction 11 over 2 End Fraction times Start Fraction 3 over 4 End Fraction 12×32×34 Start Fraction 1 over 2 End Fraction times Start Fraction 3 over 2 End Fraction times Start Fraction 3 over 4 End Fraction 12×12×34 Start Fraction 1 over 2 End Fraction times Start Fraction 1 over 2 End Fraction times Start Fraction 3 over 4 End Fraction 12×32×32
Answers
GPT-4o mini
To find the volume of a right rectangular prism, you multiply its length, width, and height. In this case, the dimensions are:
- Length = 12
- Width = 112
- Height = 34
Let's calculate the volume:
\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the values:
\[
\text{Volume} = 12 \times 112 \times 34
\]
Now, let's calculate it step by step.
1. Calculate \(12 \times 112\):
\[
12 \times 112 = 1344
\]
2. Now, multiply that result by 34:
\[
1344 \times 34 = 45696
\]
So, the volume of the prism is 45696 cubic units.
Now let's verify the volume calculated by the packing method.
You mentioned that you pack the prism with unit cubes of edge length 14 inches. First, we calculate the volume of each cube:
\[
\text{Volume of each cube} = 14 \times 14 \times 14 = 2744
\]
Then, the number of cubes packed in the prism is 36, so:
\[
\text{Total Volume from cubes} = 36 \times 2744 = 98784
\]
Now, comparing both volumes:
- Volume from edge lengths of the prism: 45696 cubic units.
- Total Volume from packed cubes: 98784 cubic units.
It looks like there's a discrepancy in the volumes calculated. The method using the edge lengths of the prism (12 x 112 x 34) gives 45696 cubic units, while the method using the cubes gives a different result. Thus, there must have been a misunderstanding or miscalculation in the setup or context of the problem.
It seems that the cube dimensions or prism dimensions should be double-checked for accuracy or additional context.
- Length = 12
- Width = 112
- Height = 34
Let's calculate the volume:
\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the values:
\[
\text{Volume} = 12 \times 112 \times 34
\]
Now, let's calculate it step by step.
1. Calculate \(12 \times 112\):
\[
12 \times 112 = 1344
\]
2. Now, multiply that result by 34:
\[
1344 \times 34 = 45696
\]
So, the volume of the prism is 45696 cubic units.
Now let's verify the volume calculated by the packing method.
You mentioned that you pack the prism with unit cubes of edge length 14 inches. First, we calculate the volume of each cube:
\[
\text{Volume of each cube} = 14 \times 14 \times 14 = 2744
\]
Then, the number of cubes packed in the prism is 36, so:
\[
\text{Total Volume from cubes} = 36 \times 2744 = 98784
\]
Now, comparing both volumes:
- Volume from edge lengths of the prism: 45696 cubic units.
- Total Volume from packed cubes: 98784 cubic units.
It looks like there's a discrepancy in the volumes calculated. The method using the edge lengths of the prism (12 x 112 x 34) gives 45696 cubic units, while the method using the cubes gives a different result. Thus, there must have been a misunderstanding or miscalculation in the setup or context of the problem.
It seems that the cube dimensions or prism dimensions should be double-checked for accuracy or additional context.