A man had a 10-gallon keg full of wine, and an empty jug. On Monday he drew off a jugful of wine and filled up the keg with water. On Friday, after the wine and water had been thorougly mixed, he drew off another jugful and again filled up the keg with water. The keg then contained equal quantities of wine and water.

What was the capacity of the jug?

if you want to know the capacity of the jug required toe bring the keg down to a 50/50 mix after n draw/refills then you can find that compacity like this

if x is the capacity of the jug then the amount of wine in the keg after the kth draw and refill is given by
10(1-(x/10))^k
now after the nth draw we want this to be equal to 5 thus
10(1-(x/10))^n=5
(1-(x/10))^n=1/2
1-(x/10)=+-(1/2)^(1/n)
x/10=1-(1/2)^(1/n)
x=10(1-(1/2)^(1/n))

thus putting in various values of n we get

<table x:str="" style="border-collapse: collapse; width: 99pt;" border="0" cellpadding="0" cellspacing="0" width="132"><col style="width: 48pt;" width="64"> <col style="width: 51pt;" width="68"> <tbody><tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt; width: 48pt;" height="17" width="64">n</td> <td class="xl22" style="width: 51pt;" width="68">x</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">1</td> <td class="xl22" x:num="" x:fmla="=10*(1-(1/2)^(1/A2))">5</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">2</td> <td class="xl22" x:num="2.9289321881345245" x:fmla="=10*(1-(1/2)^(1/A3))">2.9289322</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">3</td> <td class="xl22" x:num="2.0629947401590023" x:fmla="=10*(1-(1/2)^(1/A4))">2.0629947</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">4</td> <td class="xl22" x:num="1.591035847462855" x:fmla="=10*(1-(1/2)^(1/A5))">1.5910358</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">5</td> <td class="xl22" x:num="1.2944943670387588" x:fmla="=10*(1-(1/2)^(1/A6))">1.2944944</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">6</td> <td class="xl22" x:num="1.0910128185966073" x:fmla="=10*(1-(1/2)^(1/A7))">1.0910128</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">7</td> <td class="xl22" x:num="0.94276335736093286" x:fmla="=10*(1-(1/2)^(1/A8))">0.9427634</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">8</td> <td class="xl22" x:num="0.82995956795328785" x:fmla="=10*(1-(1/2)^(1/A9))">0.8299596</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">9</td> <td class="xl22" x:num="0.74125287712709542" x:fmla="=10*(1-(1/2)^(1/A10))">0.7412529</td> </tr> <tr style="height: 12.75pt;" height="17"> <td class="xl22" style="height: 12.75pt;" x:num="" height="17">10</td> <td class="xl22" x:num="0.6696700846319259" x:fmla="=10*(1-(1/2)^(1/A11))">0.6696701</td> </tr> </tbody></table>

Posted by Daniel on 2006-07-09 15:22:58