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A man stuck in a small sailboat on a perfectly calm lake throws a stone overboard. It sinks to the bottom of the lake.
When the water again settles to a perfect calm, is the water level in the lake higher, lower, or in the same place compared to where it was before the stone was cast in?
Did you think it will rise?
But why water level was affected?
Do you recall what does Archimedes Principle state? For an object to float on water, it has to displace that much volume of water whose weight is equal to weight of the object itself. Now if object has less density than water then obviously it has to displace lesser amount of volume of water to float on it. That means it has to sink less in water.
For a moment, let's assume the stone has very high density & hence having weight equal to hundreds of kilograms despite of having small volume.
Here, stone sinks to the bottom of the lake suggests that it is has more density than water. It can't displace the water whose weight is equal to it's weight.But when it was on sailboat it could push the sailboat down so that more water is displaced weighing equal to it's weight. Result of this, the sailboat sinks little 'deeper' compared to when stone wasn't there.
Obviously, the volume of displaced water when stone was in sailboat (due to stone only) must be greater that the volume of displaced water when stone sinks to the bottom of the lake. That's why both sailboat and stone together could float on the water. In short, sailboat helped stone to displace amount of water needed to float which results in rise in shoreline.
And when stone is thrown out of the sailboat, then ideally it can't displace more water than when it was on sailboat. Now, sailboat sinks less 'deeper' in water displacing only water need to float itself.
That's why the water level must be dropped compared to earlier. The little rise due to water displaced by stone can't exceed the earlier water level for the reason explained above.
A square tabletop measures 3n × 3n. Each unit square
is either red or blue. Each red square that doesn’t lie at the edge of
the table has exactly five blue squares among its eight neighbors. Each
blue square that doesn’t lie at the edge of the table has exactly four
red squares among its eight neighbors. How many squares of each color
make up the tabletop?
Here is correct way to count those!
How squares are arranged?
The tabletop measures 3n × 3n, so we can divide it evenly into n2 ( 3 × 3) squares that together tile the surface completely.
Let's consider a piece of square of size 3 x 3. For each such unit of 3 x 3 -
1. If the center of the square is red square, then there are 5 blue squares and 3 red squares surrounded with it.
2. If the center is blue square, then there are 4 blue and 4 red squares surrounding that square.
In any case, for 3 x 3 = 9 squares, there are 5 blue and 4 red squares.
Therefore, for tabletop of 3n x 3n, there will be 5n2 blue squares and 4n2 red squares.
