The radius of a sphere is increasing at a rate of 0.5 centimeters per minute. At a certain instant, the radius is 17 centimeters. What is the rate of change of the volume of the sphere at that instant (in cubic centimeters per minute)? Choose 1 answer: (A) (1)/(6)pi (B) 289 pi (C) 578 pi (D) 3275 pi The volume of a sphere with radius r is (4)/(3)pir^(3).

Question

The radius of a sphere is increasing at a rate of 0.5 centimeters per minute. At a certain instant, the radius is 17 centimeters. What is the rate of change of the volume of the sphere at that instant (in cubic centimeters per minute)? Choose 1 answer: (A)  (1)/(6)pi (B)  289 pi (C)  578 pi (D)  3275 pi The volume of a sphere with radius  r is  (4)/(3)pir^(3).

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Volume Formula: The formula for the volume of a sphere is V = (4/3)πr^3. We need to find dV/dt, the rate of change of volume with respect to time.Differentiate V: First, differentiate V = (4/3)πr^3 with respect to r to get dV/dr. dV/dr = 4πr^2.Chain Rule: Now, use the chain rule to find dV/dt. dV/dt = dV/dr * dr/dt. We know dr/dt = 0.5 cm/min.Substitute Values: Substitute r = 17 cm and dr/dt = 0.5 cm/min into dV/dt = 4πr^2 * dr/dt. dV/dt = 4π(17)^2 * 0.5.Calculate dV/dt: Calculate the value of dV/dt. dV/dt = 4π(289) * 0.5 = 4π * 289 * 0.5 = 578π.Final Answer: The rate of change of the volume of the sphere at that instant is 578π cubic centimeters per minute, which corresponds to answer choice (C).

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