The edge of a cube office is decreasing at a constant rate of two cm per seconds .find the rate of change of its volume at that instant when the volume is 64m cube
v = x^3
when v=64, x=4
dv/dt = 3x^2 dx/dt
now just plug in your numbers.
Watch the units.
To find the rate of change of the volume of a cube at a given instant, we can use the formula for the volume of a cube:
V = s^3
where V is the volume and s is the length of one side of the cube.
Given that the edge of the cube is decreasing at a constant rate of 2 cm per second, we can determine the rate of change of the volume when the volume is 64 m^3.
First, let's convert the volume from cubic meters to cubic centimeters:
64 m^3 = 64 * (100 cm)^3 = 64 * 1,000,000 cm^3 = 64,000,000 cm^3
Now, let's differentiate the volume formula with respect to time (t):
dV/dt = 3s^2 * ds/dt
where dV/dt is the rate of change of the volume with respect to time, ds/dt is the rate of change of the side length with respect to time, and s is the side length of the cube.
Given ds/dt = -2 cm/s (since the side length is decreasing), and the volume V = 64,000,000 cm^3, we can substitute these values into the equation to find dV/dt:
dV/dt = 3s^2 * ds/dt
dV/dt = 3(64,000,000 cm^3) * (-2 cm/s)
dV/dt = -384,000,000 cm^3/s
Therefore, the rate of change of the volume at the instant when the volume is 64 m^3 is -384,000,000 cm^3/s.