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 Rising water levels (Posted on 2006-06-26)
You have rectangular pot of water 10cm by 10cm at the bottom and 6 cm deep. It is filled to a depth of 3cm.

You have 7 solid steel shapes in front of you. The question is to find the new level of the water after each shape is put in the pot in the orientation described. The previous shape is removed before adding the next.

1. A cube 5cm on a side.

2. A prism in the shape of a right triangle with legs 5cm long. It is 5cm high but is placed on one of its legs with its hypotenuse sloping out of the water.

3. Another prism, this one having an equilateral triangle of sides 5cm and height 5cm. It is to be placed on its side with two faces sloping up out of the water.

4. A regular hexagonal prism. Each edge of the hexagon is 4cm and the height is 5cm. It is to be placed on its side.

5. A right square pyramid. Its base is 6cm on a side. It is 5cm high. It is to be placed base down.

6. A right cylinder of radius 3cm and length 5cm. It is to be placed on its side.

7a. A right cone of radius 3cm and height 5cm. It is to be placed base down.

7b. The same cone as 7a. This time placed on its side.

 No Solution Yet Submitted by Jer Rating: 4.1667 (6 votes)

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 Excellent problem! #2 solved. | Comment 7 of 26 |

For #2, divide the floor surface area into four quadrants, each 5cm by 5cm.  Three will be covered by water, the fourth by water and the right triangle "prism".

Let the height of the water be denoted by the letter h.

The volume of water above the first three quadrants will be 5 x 5 x h cm3, for a total of 75h cm3.  The volume of water above the fourth quadrant is 5cm x h2cm2/2= 2.5h2 cm3 (The slope of the wall of the prism is 1/2.)

So, we end up with a quadratic equation: 0.5h2 + 75h = 300.

Solving for h, we get h = 3.574 cm3.

Edited on June 28, 2006, 1:14 am
 Posted by Mindrod on 2006-06-28 01:01:40

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