Answer :
We can determine the parts per million (ppm) for a given error by comparing the time error to the total time over which it occurs and then multiplying the ratio by [tex]$10^6$[/tex]. We will work out the two cases separately.
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1. For an error of 1 minute every hour:
- One minute is equal to [tex]$60$[/tex] seconds.
- One hour contains [tex]$3600$[/tex] seconds.
Thus, the ratio of error time to total time is
[tex]$$
\frac{60}{3600} = \frac{1}{60}.
$$[/tex]
To convert this ratio into parts per million, multiply by [tex]$10^6$[/tex]:
[tex]$$
\text{ppm} = \frac{1}{60} \times 10^6 \approx 16666.67.
$$[/tex]
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2. For an error of 1 second every 11 days:
- One day has [tex]$86400$[/tex] seconds.
- Therefore, [tex]$11$[/tex] days have
[tex]$$
11 \times 86400 = 950400 \text{ seconds}.
$$[/tex]
The ratio of error time to total time is
[tex]$$
\frac{1}{950400}.
$$[/tex]
Multiplying this ratio by [tex]$10^6$[/tex] gives
[tex]$$
\text{ppm} = \frac{1}{950400} \times 10^6 \approx 1.05219.
$$[/tex]
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Thus, the results are:
- An error of 1 minute every hour corresponds to approximately [tex]$\boxed{16666.67\ \text{ppm}}$[/tex].
- An error of 1 second every 11 days corresponds to approximately [tex]$\boxed{1.05219\ \text{ppm}}$[/tex].
────────────────────────────
1. For an error of 1 minute every hour:
- One minute is equal to [tex]$60$[/tex] seconds.
- One hour contains [tex]$3600$[/tex] seconds.
Thus, the ratio of error time to total time is
[tex]$$
\frac{60}{3600} = \frac{1}{60}.
$$[/tex]
To convert this ratio into parts per million, multiply by [tex]$10^6$[/tex]:
[tex]$$
\text{ppm} = \frac{1}{60} \times 10^6 \approx 16666.67.
$$[/tex]
────────────────────────────
2. For an error of 1 second every 11 days:
- One day has [tex]$86400$[/tex] seconds.
- Therefore, [tex]$11$[/tex] days have
[tex]$$
11 \times 86400 = 950400 \text{ seconds}.
$$[/tex]
The ratio of error time to total time is
[tex]$$
\frac{1}{950400}.
$$[/tex]
Multiplying this ratio by [tex]$10^6$[/tex] gives
[tex]$$
\text{ppm} = \frac{1}{950400} \times 10^6 \approx 1.05219.
$$[/tex]
────────────────────────────
Thus, the results are:
- An error of 1 minute every hour corresponds to approximately [tex]$\boxed{16666.67\ \text{ppm}}$[/tex].
- An error of 1 second every 11 days corresponds to approximately [tex]$\boxed{1.05219\ \text{ppm}}$[/tex].
