An ice cube is melting so that its edge length x is decreasing at the rate of 0.1 meters per second. How fast is the volume decreasing when x = 2 meters?
To find how fast the volume is decreasing when the edge length of the ice cube is 2 meters, we need to use the chain rule from calculus.
The volume of a cube is given by V = x^3, where x represents the edge length. We want to find how fast the volume (V) is changing with respect to time (t). Let's denote the rate at which the edge length is changing with dx/dt and the rate at which the volume is changing with dV/dt.
Given dx/dt = -0.1 m/s (negative sign indicates that the edge length is decreasing), we need to find dV/dt when x = 2 m.
Using the chain rule, we have:
dV/dt = dV/dx * dx/dt
We know that dV/dx is the derivative of the volume with respect to the edge length, which is 3x^2 (dV/dx = 3x^2).
Substituting the given values, we have:
dV/dt = (3x^2) * (-0.1)
Plugging x = 2 into the equation:
dV/dt = (3(2^2)) * (-0.1)
= (3 * 4) * (-0.1)
= 12 * (-0.1)
= -1.2 m^3/s
Therefore, when the edge length is 2 meters, the volume of the ice cube is decreasing at a rate of -1.2 cubic meters per second.
V = x^3
dV/dt = 3x^2(dx/dt)
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