Answer:
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 <em>between</em> 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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<em>More precise discussion of the cutoff</em>
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.
