Two Jars

 

Solution

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Both jars are the circular cylinders. The volume of a circular cylinder is πr2h, where r is the radius of the bases, and h is the perpendicular distance between the planes that contain the bases. In our case h is the height of a jar. Since the volume of each jar is identical, but their heights are different, then, obviously the radii of their respective bases are also different. Thus, the bases' areas of each jar are:
higher jar: πr2 = 6/9.
lower jar: πr2 = 6/4.

Let's multiply both decimals by 6:
higher jar: 6/9 x 6 = 36/9 = 4 (the area of the higher jar's base in square units).
lower jar: 6/4 x 6 = 36/4 = 9 (the area of the lower jar's base in square units).

Since we've multiplied both decimals by 6 we can describe the volumes of the jars as 36 cubic units each. That means 1 liter equals 6 cubic units. In order to measure 4 liters we have to get 24 cubic units of water in one of the jars.

Step 1. Fill out both jars - 36 cubis units of water in each.

Step 2. Place the higher jar into the lower. Such a procedure will displace a certain volume of water from the lower jar. Ignoring the thikness of the jar's glass, the volume left in the lower jar will be equal the product of the jar's height (i.e. 4) and the difference between the areas of both jars' bases (i.e. 9-4). In other words the volume of water left in the lower jar equals 4 x (9-4) = 4 x 5 = 20 cubic units.

Step 3. Get the higher jar out of the lower one.

Step 4. Pour the water from the higher jar into the lower one until the latter is brimful again. Since the lower jar contains 20 cubic units of water already, the 16 cubic units will be poured into it from the higher jar. Now there are 20 cubic units of water in the higher jar and 36 - in the lower.

Step 5. Place the higher jar once again into the lower one. The procedure like in the step 2 will leave in the lower jar 20 cubic units, displacing 16 cubic units of water.

Step 6. Get the higher jar out of the lower one.

Step 7. Pour the water from the higher jar into the lower one until the latter is brimful again. Like in step 4 this will add 16 cubic units to the lower jar and thus, 4 cubic units are left now in the higher jar.

Step 8. Place the higher jar once again into the lower one. The procedure like in the steps 2 and 5 will leave in the lower jar 20 cubic units, displacing 16 cubic units of water from it.

Step 9. Once again get the higher jar out of the lower one. Now the total volume of the water in both jars is 24 cubic units - 4 in the higher and 20 in the lower one.

Step 10. Pour all the water either from the higher one into the lower or vice versa - from the lower into the higher. Now one of the jars holds the volume of the water which has been sought for, i.e. 24 cubic units or 4 liters.

Posted: December 7, 2007
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