The surface area of a sphere is increasing at a rate of 14 pi square meters per hour. At a certain instant, the surface area is 36 pi square meters. What is the rate of change of the volume of the sphere at that instant (in cubic meters per hour)? Choose 1 answer: (A) 36 pi (B) (sqrt(14 pi))^(3) (C) (7)/(12) (D) 21 pi The surface area of a sphere with radius r is 4pir^(2). The volume of a sphere with radius r is (4)/(3)pir^(3).

Question

The surface area of a sphere is increasing at a rate of  14 pi square meters per hour. At a certain instant, the surface area is  36 pi square meters. What is the rate of change of the volume of the sphere at that instant (in cubic meters per hour)? Choose 1 answer: (A)  36 pi (B)  (sqrt(14 pi))^(3) (C)  (7)/(12) (D)  21 pi The surface area of a sphere with radius  r is  4pir^(2). The volume of a sphere with radius  r is  (4)/(3)pir^(3).

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Surface Area Calculation: Given the surface area S = 36 pi square meters, we can find the radius r of the sphere using the formula for surface area S = 4 pi r^2.Radius Calculation: Solve for r: 36 pi = 4 pi r^2. Divide both sides by 4 pi: r^2 = 36 pi / 4 pi = 9. Take the square root of both sides: r = 3 meters.Rate of Change Relation: The rate of change of the surface area is given as dS/dt = 14 pi square meters per hour. We can relate this to the rate of change of the volume dV/dt using the chain rule in calculus: dV/dt = dV/dr * dr/dS * dS/dt.Volume Differentiation: First, find dV/dr using the volume formula V = (4/3) pi r^3. Differentiate V with respect to r: dV/dr = 4 pi r^2.Surface Area Differentiation: Next, find dr/dS using the surface area formula S = 4 pi r^2. Differentiate S with respect to r: dS/dr = 8 pi r. Now, solve for dr/dS: dr/dS = 1 / (dS/dr) = 1 / (8 pi r).dr/dS Calculation: Substitute r = 3 meters into dr/dS: dr/dS = 1 / (8 pi * 3) = 1 / (24 pi).Substitute r into dr/dS: Now, substitute dV/dr = 4 pi r^2 and dr/dS = 1 / (24 pi) into the chain rule equation. dV/dt = (4 pi r^2) * (1 / (24 pi)) * (14 pi).Substitute into Chain Rule: Simplify the equation: dV/dt = (4 pi * 9) * (1 / (24 pi)) * (14 pi). dV/dt = (36 pi) * (1 / (24 pi)) * (14 pi).Simplify Chain Rule: Cancel out the pi terms and simplify: dV/dt = (36 / 24) * 14. dV/dt = (3 / 2) * 14. dV/dt = 21.

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