A technical machinist is asked to build a cubical steel tank that will hold $75 \mathrm{~L}$ of water.
Calculate in meters the smallest possible inside length of the tank. Round your answer to the nearest $0.01 \mathrm{~m}$.

Step 1 ：A technical machinist is asked to build a cubical steel tank that will hold 75 L of water. We are asked to calculate in meters the smallest possible inside length of the tank. We are to round our answer to the nearest 0.01 m.

Step 2 ：The volume of a cube is given by the formula \(V = a^3\) where \(a\) is the length of one side of the cube. We know that the volume of the tank is 75 L, which is equivalent to 0.075 m^3 (since 1 L = 0.001 m^3).

Step 3 ：We can solve for \(a\) by taking the cube root of the volume. So, \(a = \sqrt[3]{V}\).

Step 4 ：Substituting the given volume into the equation, we get \(a = \sqrt[3]{0.075}\).

Step 5 ：Calculating the cube root of 0.075, we get \(a = 0.42\).

Step 6 ：Final Answer: The smallest possible inside length of the tank is \(\boxed{0.42 \mathrm{~m}}\).