The side of a cube is decreasing at a rate of 9 millimeters per minute.At a certain instant, the side is 19 millimeters.What is the rate of change of the volume of the cube at that instant (in cubic millimeters per minute)?Choose 1 answer:(A) −3249(B) −6859(C) −9747(D) −729
Q. The side of a cube is decreasing at a rate of 9 millimeters per minute.At a certain instant, the side is 19 millimeters.What is the rate of change of the volume of the cube at that instant (in cubic millimeters per minute)?Choose 1 answer:(A) −3249(B) −6859(C) −9747(D) −729
Volume Formula Derivation: The volume of a cube is given by the formula V=s3, where s is the side length of the cube. To find the rate of change of the volume, we need to differentiate the volume with respect to time.
Rate of Change Calculation: So, dtdV=3s2⋅dtds. Here, dtds is the rate of change of the side length, which is −9mm/min (negative because the side is decreasing).
Substitute Values: Now we plug in the values: s=19mm and dtds=−9mm/min. So, dtdV=3×(19)2×(−9).
Final Calculation: Calculating that out, dtdV=3×361×(−9)=3×(−3249)=−9747 cubic millimeters per minute.
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