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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! #4 solved | Comment 9 of 26 |

Instead of laying out a full solution here, I'll simply describe the method:

The volume of water below the horizontal centerline of the hexagonal prism is equal the volume of water in a 10 x 10 x h' tank minus half of the volume of the hexagonal prism, where h' is half the height of the hexagonal prism.  Subtract this volume from 300 cubic centimeters and that's how much water lies above the centerline of the hexagonal prism.  The upper half of the hexagonal prism displaces water in the same fashion as the equilateral triangle prism of problem #3 to a height of h" cubic centimeters, but with an additional 4 x 5 x h" cubic centimeters displaced by the rectangular solid portion between the two sloped sides.  Solve another quadratic equation, and you have the height of water displaced above the hexagonal prism's centerline.  Add this to half the height of the hexagonal prism and you have your answer.

The original answer I got, resulted from a calculation error.  After going over my work again, I found h = 4.32821 cm.

Edited on July 1, 2006, 3:36 pm
 Posted by Mindrod on 2006-06-28 01:59:15

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