doc1 - BUIMPLE 1 Air is being pumped into a spherical...

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Unformatted text preview: BUIMPLE 1 Air is being pumped into a spherical bailoon so that its volume increasa at a rate of EU omzfs. How fast is the radius of the balloon increasing when the diarneter is 40 Cm? SOLUTION We start by identifying two things: the given infunnau-on: the rate of increase of thevciurrie cf air is 50 :mzfs and the unknown: the rate of increase of the radius when the diameter is 4D cm In order to express thae quantities mathematioaily, we introduce some suggestive notation: Let V be the volume of the balioan and iet r be its radius. The key thing to rernernher is that rates af change are derivatives. In this prehiern, the vaiurne and the radius are both functions at the tirne. The rate at increase at the volume with rapect to tirne is the derivative ”WW and the rate at increase of the radius is d’rdt. We can therefare restate the given and the unknown as faiicwsi dV Given: = so cmafs dt dr Unknown: when r = Cm dt In arder to connect ”War and ”7dr, we first relate Vand r by the formula for the vaiurne of a sphere: [n arder ta use the given infdrrnatian, we differentiate each side of this eduatiah with respect to t. To differentiate the ridht side, we need to use the chain Rule: dl/ dv dr l— dr dt dr dt 7 dt Now we sdive far the unknnwn quantity: dr dt dt If we putr : 20 and W’at : so in this eduatiah, we obtain |— air 4nl—I [— The radius cf the baHoon is increasing at the rate of 50"E4011] a 0.0099 cmr's. dV dr 1 ...
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