Answer:
1/27 m/s
Explanation:
To address the problem of determining the rate at which the side of a cube changes given that its volume increases at a rate of 4 m³/s, we need to apply the concept of related rates in calculus. This involves using derivatives to relate the rates of change of different quantities.
Given:
We know the volume of a cube is:

Where 'L' is the length of one side of the cube. Take the derivative of this function with respect to time, 't'.
![\Longrightarrow (d)/(dt)[V] = (d)/(dt)[L^3]\\\\\\\\\Longrightarrow (dV)/(dt)= 3L^2\cdot(dL)/(dt)](https://img.qammunity.org/2024/formulas/mathematics/college/k1jwegeajy67cjm2vojoswmz3ig5n6igzp.png)
Solve for dL/dt:

Plug in our given values:

Therefore, the rate at which the side of the cube changes when its length is 6 m is 1/27 m/s.