Using linear approximations, if a spherical ballon has a radius of 5 cm, find the increase in volume of the balloon when the radius expands by 0.02 cm

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Let volume of the sphere of radius r be V

Thus

`V=(4/3)pir^3`

`(dV)/(dr)=4pir^2`

`But`

`DeltaV=(dV)/(dr)Deltar`

`` because

`lim_(Deltar->0)(DeltaV)/(Deltar)=(dV)/(dr)`

`Thus`

`DeltaV=(4pir^2)Deltar`

`DeltaV=` Change in volume

`Deltar=`  Change in radius

Thus

`r=5`

`Deltar=.02`

`DeltaV=(4xx3.14xx5^2)xx.02`

`DeltaV=314xx.02`

`DeltaV=0.0628`  cubic cm

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