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Knowledge of weighing objects with scales.
Let's first study the situation that only one side plate of the balance is allowed to put weights, and it is required to weigh items at one time.
For example, if you put a weight on one side of the balance, you have to weigh everything from 1g to 30g at one time. What weight should you at least have?
If you want to weigh at one time, the number of weights should be "less" and the grams of each weight should not be the same. If you can piece together several weights to be weighed, try to put them together.
Obviously, the weights of 1g and 2g are indispensable. 1+2 = 3 (grams), and the weight of 3 grams can be omitted. Using weights of 1 g and 2 g, you can't weigh 4 g at a time, but you must have a weight of 4 g. The weight is 4g, and the weight is 1 g and 2g, which can be called 5g, 6g and 7g respectively. According to this idea, we simulated the situation of weighing things on the balance and made the following table:
Placement weight (gram)
Weigh the goods (grams)
1
1
2
2
3+ 1
three
four
four
4+ 1
five
4+2
six
4+2+ 1
seven
eight
eight
……
……
8+4+2+ 1
15
16
16
……
……
16+8+4+2
30
16+8+4+2+ 1
3 1
As can be seen from the table, when the weight is 30g, four weights are used. However, when weighing 1g to 30g, five weights should be prepared, namely 1g, 2g, 4g, 8g and 16g, and the maximum weighing weight using these five weights is 1+2+4+8+65438+.
Let's first find out the relationship between these five weights, L grams, 2 grams, 4 grams, 8 grams, 16 grams, in the order from light to heavy. It is not difficult to find that the weight of two adjacent weights is twice that of the lighter weight. Therefore, the weight can only be placed on one side plate of the balance, and it is required to weigh 1g to several kilograms of whole grams at one time. At least the weight of each weight to be prepared is 1g, and the rest can be obtained by "twice method" in turn.
Intensive store knowledge
The shape of floor tiles is often square and rectangular, and we have also seen regular hexagonal floor tiles. Whether it is a square, rectangular or hexagonal floor tile, the middle of a piece of ground can be paved without gaps or overlaps, that is, densely paved. What other shapes of graphics can be densely laid on the ground? When students think about this problem, they always experiment with the help of the pictures they draw and draw conclusions through actual observation.
In fact, the life problem of floor tile paving also has mathematical truth, which can be solved by theoretical analysis with the knowledge that the circle angle learned from mathematics is 36 degrees.
As we all know, when paving the floor, the floor should be covered, and there should be no gaps between the floor tiles. If the floor tile used is square, and each corner of it is a right angle, then four squares are put together, and the four corners of the common vertex just spell a 36-degree fillet. Every corner of a regular hexagon is 120 degrees.
When three regular hexagons are put together, the sum of the three angles on the common vertex is exactly 36o degrees. Besides squares and rectangles, regular triangles can also be densely laid on the ground. Because every internal angle of a regular triangle is 60 degrees, when six regular triangles are put together, the sum of the degrees of the six angles at the vertices of the common triangle is exactly 36 degrees.
It is precisely because the sum of several angles on the vertices of a square and a regular hexagon is exactly 36 degrees that the ground can be paved densely and beautifully.
What other shapes of graphics can be densely laid on the ground? Will you answer this question from a mathematical point of view now? Try it?