The volume of a sphere is increasing at a rate of 25π cubic meters per hour.At a certain instant, the volume is 332π cubic meters.What is the rate of change of the surface area of the sphere at that instant (in square meters per hour)?Choose 1 answer:(A) 932π(B) 25π(C) 1625(D) (325π)2The surface area of a sphere with radius r is 4πr2.The volume of a sphere with radius r is 34πr3.
Q. The volume of a sphere is increasing at a rate of 25π cubic meters per hour.At a certain instant, the volume is 332π cubic meters.What is the rate of change of the surface area of the sphere at that instant (in square meters per hour)?Choose 1 answer:(A) 932π(B) 25π(C) 1625(D) (325π)2The surface area of a sphere with radius r is 4πr2.The volume of a sphere with radius r is 34πr3.
Given Information: Given: The volume of a sphere is increasing at a rate of 25π cubic meters per hour.Volume of a sphere formula: V=34πr3.Differentiate both sides with respect to time t to find the rate of change of the radius.dtdV=4πr2dtdr.
Differentiate Volume Formula: Substitute the given rate of change of volume into the differentiated volume formula.25π=4πr2dtdr.Solve for dtdr.dtdr=4πr225π.
Substitute Rate of Change: Given: The volume at a certain instant is (32π)/3 cubic meters.Use the volume formula to find the radius at that instant.(32π)/3=(4/3)πr3.Solve for r3.r3=(32π)/(4π).r3=8.
Find Radius at Instant: Find the radius by taking the cube root of r3.r=cube root of 8.r=2 meters.
Calculate Radius: Substitute r=2 into the expression for dtdr. dtdr=4π⋅2225π. dtdr=16π25π. dtdr=1625 meters per hour.
Calculate Rate of Change: Surface area of a sphere formula: A=4πr2. Differentiate both sides with respect to time t to find the rate of change of the surface area. dtdA=8πrdtdr.
Differentiate Surface Area Formula: Substitute r=2 and dtdr=1625 into the differentiated surface area formula.dtdA=8π⋅2⋅(1625).dtdA=8π⋅2⋅(1625).dtdA=16100π.dtdA=425π.
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