Final answer:
The side of a cube whose volume is increasing at 4 m³/sec changes at a rate of 4/27 m/sec when its side length is 3 m.
Step-by-step explanation:
The student's question seeks to find the rate at which the side of a cube changes when its volume is increasing at a rate of 4 m³/sec and its side length is 3 m. To solve this, we can use related rates, which is a method to determine the rate at which one quantity changes in relation to another. The volume V of a cube is given by V = s³, where s is the side length. Differentiating both sides with respect to time t, we get dV/dt = 3s² ds/dt, where dV/dt is the rate of change of the volume and ds/dt is the rate of change of the side length.
Given that dV/dt = 4 m³/sec and s = 3 m, we can substitute these values into the equation to get 4 = 3(3)² ds/dt, which simplifies to 4 = 27 ds/dt. Solving for ds/dt gives us ds/dt = 4/27 m/sec. Therefore, the side length of the cube is changing at a rate of 4/27 m/sec.