Final answer:
To calculate the rate at which the cube's volume is changing when the edge length is 2 inches, we apply the chain rule to the relationship between surface area and volume of a cube and find that the volume is changing at a rate of 18 cubic inches per second.
Step-by-step explanation:
If a cube's surface area is increasing at a rate of 36 square inches per second, we can use the relationship between the surface area and the volume of a cube to find the rate at which the cube's volume is changing. The surface area (SA) of a cube is given by SA = 6s², where s is the length of a side. To find the rate of change of the volume, we need to use the chain rule from calculus since the volume (V) is given by V = s³.
Given that the rate of change of the surface area (dSA/dt) is 36 in²/s, we first find ds/dt, the rate of change of the side length of the cube. If SA = 6s², then dSA/dt = 12s (ds/dt). Solving for ds/dt gives us ds/dt = (dSA/dt) / (12s).
When the edge length is 2 inches, we can substitute s = 2 inches and dSA/dt = 36 in²/s into the equation to find ds/dt:
ds/dt = 36 in²/s / (12 × 2 in) = 1.5 in/s
Now we use the formula for the volume of a cube V = s³ to find dV/dt, the rate of change of the volume:
dV/dt = 3s² (ds/dt)
Substituting s = 2 inches and the value we found for ds/dt:
dV/dt = 3 × (2 in)² × 1.5 in/s = 3 × 4 in² × 1.5 in/s = 18 in³/s
Therefore, the rate at which the volume is changing when the edge length is 2 inches is 18 cubic inches per second.