Final answer:
The length of each edge of the cube is 8 inches.
Step-by-step explanation:
To find the length of each edge of the cube, we need to use the formulas for volume and surface area of a cube. Let's assume the length of each edge is 'x'. The formula for volume is V = x^3, and the formula for surface area is SA = 6x^2.
We are given that the volume is increasing at a rate of 24 in^3/min, so we have dV/dt = 24. Differentiating the volume equation with respect to time gives us dV/dt = 3x^2 * dx/dt. Therefore, we have 24 = 3x^2 * dx/dt.
We are also given that the surface area is increasing at a rate of 12 in^2/min, so we have dSA/dt = 12. Differentiating the surface area equation with respect to time gives us dSA/dt = 12x * dx/dt. Therefore, we have 12 = 12x * dx/dt.
Simplifying the equations, we have x^2 * dx/dt = 8 and x * dx/dt = 1. Dividing the two equations gives us x = 8/1 = 8. Therefore, the length of each edge of the cube is 8 inches.