An ice cube is melting, and the lengths of its sides are decreasing at a rate of 0.4 millimeters per minute.
At what rate is the volume of the ice cube decreasing when the lengths of the sides of the cube are equal to 15 millimeters?
Give your answer correct to the nearest cubic millimeter per minute.
v = s^3
dv/dt = 3 s^2 ds/dt
here ds/dt = -0.4 and s = 15
An ice cube is melting at a rate of 0.2 cm/min. side length of cube is 3 cm. What is rate of change of volume after 5 mins
5.76
172.8******
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To find the rate at which the volume of the ice cube is decreasing, 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 a side.
First, let's differentiate both sides of the equation with respect to time t:
dV/dt = d/dt (s^3)
Next, we need to use the chain rule to differentiate the right-hand side of the equation:
dV/dt = 3s^2 * ds/dt
Given that the lengths of the sides of the cube are decreasing at a rate of 0.4 millimeters per minute, we have ds/dt = -0.4 mm/min. Since the length of the side is given as 15 mm, we have s = 15 mm.
Now, substitute the values into the equation:
dV/dt = 3(15^2) * (-0.4)
Simplifying the equation:
dV/dt = 3(225) * (-0.4)
dV/dt = -270 cubic millimeters per minute
Therefore, the rate at which the volume of the ice cube is decreasing, when the lengths of the sides of the cube are equal to 15 millimeters, is approximately -270 cubic millimeters per minute.