calc project .pdf - Carrie Yung AP Calculus Mr Lord Project...

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Carrie YungAP CalculusMr. LordProject 1Can Cans Be Optimized?A.Prediction - I think that the salt can will be more “volume-optimized” because the cylinder is bigger than the corn can, the salt can also has a larger surface area than the corn can.B.C.SALT CAN:Hight:15.5cmRadius:3.75cmVolume:==Surface area: SA=SA===V=πr2hV=π(3.752)(15.5) 218π684.9cm32πr2+ 2πrh2π(3.752) + 2π(3.75)(15.5)116.25π+ 28.125π114.375πcm2CORN CANHight: 11cmRadius:3.65Volume:==SA=SA===V=πr2hV=π(3.652)(11) 146.5π460.4cm32πr2+ 2πrh2π(3.652) + 2π(3.65)(11)26.645π+ 80.3π106.945πcm2SALT CANPrimary: Secondary: SA=Actual:===PLUG INTO PRIMARY EQUATIONRelative extrema occur when the derivative is zero r=4.4cm (rejected negative) h=8.6cm MAX VOLUME =R=πr2h2πr2+ 2πrh114.375π114.375π2πr2+ 2πrh114.375π/2π2π(rh+r2)/2π57.1875/r r(h+r)/r(57.1875/r)r=hR=πr2[(57.1875/r)r]R= 57.1875πrπr3R= 57.1875π3πr20 = 57.1875π3πr2(57.1875π)/(3π) = (3πr2)/(3π)19.0625 =r2(57.1875/r)r=h(57.1875/4.4)4.4 =hV=πr2hV=π(4.42)(8.6) 166.496πV= 523.1cm3CORN CAN Primary: Secondary: SA=Actual:=PLUG INTO PRIMARY EQUATION Relative extrema occur when the derivative is zeror=4.2cm (rejected negative)

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