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A reservoir has the shape of a truncated cone as shown in the ï¬gure above. It measuresÂ 3Â m high. The radius of the base isÂ 2Â m and the top radius isÂ 4Â m. It is ï¬lled with water. We want to calculate the amount of work required to pump all of its water through the hose standingÂ 2Â m above the reservoir.
(i) LetÂ xÂ Â be the height in meters measured from the base of the reservoir. The weight of a thin water layer between heightsÂ xÂ Â andÂ x+Î”xÂ Â is approximatelyÂ P(x)Î”xÂ . What isÂ P(x)Â ?
P(x)=Â Â Â Â
Express the answer using a formula. Recall that the density of water isÂ Ï=1000Kg/m3Â Â and the acceleration due to gravity on the earthâ€™s surface isÂ g=9.8m/s2Â .
(ii) The work in Joules required to pump the thin water layer toÂ 2Â m above the reservoir is approximatelyÂ w(x)Î”xÂ . What isÂ w(x)Â ?
w(x)=Â Â Â Â
Express the answer as a formula.
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