The radius of the base of a cylinder is increasing at a rate of 7 millimeters per hour. The height of the cylinder is fixed at 1.5 millimeters. At a certain instant, the radius is 12 millimeters. What is the rate of change of the volume of the cylinder at that instant (in cubic millimeters per hour)? Choose 1 answer: (A) 216 pi (B) 1512 pi (C) 126 pi (D) 252 pi The volume of a cylinder with radius r and height h is pir^(2)h.

Question

The radius of the base of a cylinder is increasing at a rate of 7 millimeters per hour. The height of the cylinder is fixed at 1.5 millimeters. At a certain instant, the radius is 12 millimeters. What is the rate of change of the volume of the cylinder at that instant (in cubic millimeters per hour)? Choose 1 answer: (A)  216 pi (B)  1512 pi (C)  126 pi (D)  252 pi The volume of a cylinder with radius  r and height  h is  pir^(2)h.

Bytelearn · Accepted Answer

Volume Formula Derivation: The formula for the volume of a cylinder is V = πr^2h. We need to find dV/dt, the rate of change of the volume.Differentiating Volume Formula: First, let's differentiate V with respect to t: dV/dt = d(πr^2h)/dt. Since h is constant, we can take it outside the differentiation: dV/dt = h * d(πr^2)/dt.Differentiating πr^2 with respect to r: Now differentiate πr^2 with respect to r: d(πr^2)/dr = 2πr. Then multiply by dr/dt to get d(πr^2)/dt: d(πr^2)/dt = 2πr * dr/dt.Calculating d(πr^2)/dt: We know dr/dt = 7 mm/hour (the rate at which the radius is increasing) and r = 12 mm (the radius at the instant we're interested in).Substitute Values: Plug in the values: d(πr^2)/dt = 2π * 12 mm * 7 mm/hour = 168π mm^2/hour.Calculate dV/dt: Now, multiply by the height h = 1.5 mm to find dV/dt: dV/dt = 168π mm^2/hour * 1.5 mm = 252π mm^3/hour.

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