If boiling water is poured into a thick drinking glass as well as a very thin wine glass, of the two, which is more likely to crack?
Answer: The thick glass is more likely to crack since glass is a poor conductor of heat. In a thin glass, the heat passes more quickly from the glass into the surrounding air, causing the glass to expand equally. When hot water is poured into a thick glass, the inner surface expands, but the outer surface does not. It is this extreme stress on the glass that causes it to crack.
Solution:
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The thick glass is more likely to crack since glass is a poor conductor of heat. In a thin glass, the heat passes more quickly from the glass into the surrounding air, causing the glass to expand equally. When hot water is poured into a thick glass, the inner surface expands, but the outer surface does not. It is this extreme stress on the glass that causes it to crack.
Show Answer
The thick glass is more likely to crack since glass is a poor conductor of heat. In a thin glass, the heat passes more quickly from the glass into the surrounding air, causing the glass to expand equally. When hot water is poured into a thick glass, the inner surface expands, but the outer surface does not. It is this extreme stress on the glass that causes it to crack.
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When the celebrated German mathematician Karl Friedrich Gauss (1777-1855) was nine he was asked to add all the integers from 1 through 100. He quickly added 1 to 100, 2 to 99, and so on for 50 pairs of numbers each adding to 101. Answer: 50 X 101=5,050. What is the sum of all the digits in integers from 1 through 1,000,000,000? (That’s all the digits in all the numbers, not all the numbers themselves.)
Answer: The numbers can be grouped by pairs: 999,999,999 and 0; 999,999,998 and 1' 999,999,997 and 2; and so on.... There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then: (500,000,000 X 81) + 1= 40,500,000,001.
Solution:
Show Answer
The numbers can be grouped by pairs:
999,999,999 and 0;
999,999,998 and 1′
999,999,997 and 2;
and so on….
There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then:
(500,000,000 X 81) + 1= 40,500,000,001.
Show Answer
The numbers can be grouped by pairs:
999,999,999 and 0;
999,999,998 and 1′
999,999,997 and 2;
and so on….
There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then:
(500,000,000 X 81) + 1= 40,500,000,001.
999,999,999 and 0;
999,999,998 and 1′
999,999,997 and 2;
and so on….
There are half a billion pairs, and the sum of the digits in each pair is 81. The digits in the unpaired number, 1,000,000,000, add to 1. Then:
(500,000,000 X 81) + 1= 40,500,000,001.
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