Question

A bowl is formed by revolving the lower half of the ellipse x^2 + 4(y − 1)^2 = 4 around the y-axis. (Repeat: around the Y axis.) Water is then poured in to the bowl up to a depth of 0.75 units. THEN a lead sphere of diameter 1 unit is dropped in the bowl and settles to the bottom. Does the water overflow the bowl when the sphere is dropped in? You can use a computational aid (Maple of Wolfram Alpha) to calculate integrals, but you have to be clear about what integral(s) you set up and had calculated. This is NOT a yes or no question, your response must be backed up quantitatively and specifically.

Answer 1

Answer Equalion o pe 4 4 1 4 (y-1 Cx-AJ Ch. k)Co,) Centru Senimajan axis, a=2 . Semimimor ai) = J So u plot Leof lika dinis (-4) (2,1) we have te (-2,0) (2,0) helune of hadas shado egion abut -axis. "To calcudat volume Let us take (2,1) 4of Atip of longth 4 R. -y-j ot 4 Volume elowent, dV: dV TC 4-4 y:- Lowerlimit Upper nit

dv ソ=0 1 - 4 4- -h)- Let 9-1 =大dy こdt t o- 1 Lower Linit t 1-L o Uppo Lmut 4TC 4代 4TC 870 3 8. 38units mits Hence Tatal volne of jhe bowl = 8.30unite. l 4- o.75 units: - Hho volme Now Caleulating o-75 fo.75 1-(3ay-) TC (1--) Louwe Limit, *= o-J--1

-0.2 -O.25 V= 4TC 4-T 4.15)-fo5(-). 410 + - - 5. 30 nit into duoppad s Yadius iunit Aphur ofRaad f Now Volume of Aphu - 3 4.188 = 4.19 units At oing wter w overf e Volume of watev jn bool > Volume of the bowl Volu me of sphre V Vs V 9.49 V a.38) 4.19 5.30 Hena the walen ual averfla>