Question

I NEED HELP I DON'T KNOW HOW TO DO THIS
What is the order of magnitude for the total weight of 5 cars that weigh 3,000 pounds each?

20,000 people voted in each county, and there are 7 counties. What is the order of magnitude for the total number of people who voted?
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Answer 1

Answer:

• 4
• 5

Step-by-step explanation:

For numbers 1 or greater whose leading digits are about 3.2 or less, the "order of magnitude" is the number of decimal places between the first digit and the decimal point. (If the number is more than that, the order of magnitude is 1 higher.)

a) The total weight of the 5 cars is about 5×3000 = 15,000. There are 4 digits between the leading 1 and the decimal point, so this number has an order of magnitude of 4.

Another way to find it is to take the (base-10) logarithm and round to the nearest integer.

  log(15000) ≈ 4.2 ≈ 4 . . . . the order of magnitude is 4

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b) The total number of voters is 7×20,000 = 140,000. There are 5 digits between the leading 1 and the decimal point, so the order of magnitude is 5.

  log(140,000) ≈ 5.1 ≈ 5 . . . . the order of magnitude is 5

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More precise discussion of the cutoff

The reason we put a cutoff of about 3.2 is that log(3.2) ≈ 0.5. The logarithm will actually round up if the number is about 3.162278 or above. So, the order of magnitude of 31.6 might be said to be 1, while the order of magnitude of 31.7 would be said to be 2.