As you know, a cube with each side 4 m in length has a volume of 64 m3. Each side of the
cube is now doubled in length. What is the ratio of the new volume to the old volume? Why
is this ratio not simply 2? Include a sketch with dimensions.
Because since the length is doubled, it is now 8m, if you cube that to get the volume, it'll be a ratio of 8
To find the new volume of the cube after each side is doubled in length, we need to determine the new side length first.
Given that the original cube has a side length of 4 m and a volume of 64 m³, we can use the formula for the volume of a cube to find the ratio.
Volume of a cube = (Side length)³
Substituting the values we have:
64 m³ = (4 m)³
Now, let's find the new side length of the cube:
(4 m)³ = (New side length)³
64 m³ = (New side length)³
Taking the cube root of both sides gives us:
New side length = ∛(64 m³) = 4∛4 m
Since each side length is doubled, the new side length becomes:
New side length = 2 * 4∛4 m = 8∛4 m
Now, we can calculate the new volume of the cube:
New volume of the cube = (New side length)³
Plugging in the value for the new side length:
New volume of the cube = (8∛4 m)³
Simplifying further:
New volume of the cube = 512 m³
To find the ratio of the new volume to the old volume:
Ratio = New volume of the cube / Old volume of the cube
Ratio = 512 m³ / 64 m³
Ratio = 8
Therefore, the ratio of the new volume to the old volume is 8, not 2. This is because volume is a three-dimensional measure, and when the side length doubles, the volume increases by a factor of (2³ = 8).
To find the ratio of the new volume to the old volume when each side of a cube is doubled, we can follow these steps:
Step 1: Find the old volume of the cube.
In this case, the old volume is given as 64 m^3 because it states that the cube with each side 4 m in length has a volume of 64 m^3.
Step 2: Determine the new length of each side.
Since each side of the cube is doubled in length, the new length of each side will be 4 m x 2 = 8 m.
Step 3: Calculate the new volume of the cube.
The new volume can be found by cubing the new length of each side:
New volume = (8 m) ^ 3 = 512 m^3
Step 4: Find the ratio of the new volume to the old volume.
The ratio of the new volume to the old volume can be calculated as:
Ratio = New volume / Old volume = 512 m^3 / 64 m^3 = 8
Therefore, the ratio of the new volume to the old volume is 8:1, which means the new volume is eight times larger than the old volume. It is not simply 2 because the volume of a cube increases with the cube of its side length, not linearly.
Here is a sketch to help visualize the situation:
```
Old Cube:
______
/ /|
/ / | 4 m
/______/ |
| | /
| |/
4 m
New Cube:
______________
/ /|
/ / | 8 m
/________________/ |
/ / |
/ / |
/________________/ |
| | /
| | /
| | /
| | /
4 m
```
I hope this explanation helps! Let me know if you have any further questions.