How does the surface area-to-volume ratio of
a 1-mm cube compare to the surface area-to-
volume ratio of a 3-mm cube?
1. The ratio does not change.
2. The ratio increases as the cube becomes
larger.
3. Increasing the volume increases the ra-
tio.
4. The 3 mm cube has a higher ratio.
To compare the surface area-to-volume ratio of a 1-mm cube and a 3-mm cube, we need to calculate the surface area and volume of each cube.
The surface area of a cube can be found by using the formula: SA = 6s^2, where s represents the length of one side of the cube.
The volume of a cube can be found by using the formula: V = s^3, where s represents the length of one side of the cube.
Let's calculate the surface area and volume for each cube:
For the 1-mm cube:
Surface Area = 6 * (1 mm)^2 = 6 mm^2
Volume = (1 mm)^3 = 1 mm^3
For the 3-mm cube:
Surface Area = 6 * (3 mm)^2 = 54 mm^2
Volume = (3 mm)^3 = 27 mm^3
Now, let's compare the surface area-to-volume ratios:
For the 1-mm cube:
Surface Area-to-Volume Ratio = Surface Area / Volume = 6 mm^2 / 1 mm^3 = 6 mm^-1
For the 3-mm cube:
Surface Area-to-Volume Ratio = Surface Area / Volume = 54 mm^2 / 27 mm^3 = 2 mm^-1
Therefore, the surface area-to-volume ratio of the 1-mm cube is 6 mm^-1, and the ratio for the 3-mm cube is 2 mm^-1.
From the calculations, we can see that as the cube becomes larger (from 1 mm to 3 mm), the surface area-to-volume ratio decreases. This means that the 3-mm cube has a lower ratio compared to the 1-mm cube. Therefore, option 4 is incorrect.
The correct answer is option 2. The surface area-to-volume ratio increases as the cube becomes larger is the incorrect answer.
Option 3 is incorrect as well. Increasing the volume does not necessarily increase the ratio. In this case, increasing the volume (from 1 mm^3 to 27 mm^3) has actually resulted in a decrease in the ratio.
Thus, the ratio does not remain the same, increase, or consistently change with increasing volume. The 3 mm cube has a lower ratio than the 1-mm cube.
Option 2 is the correct answer.