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Estimation quiz!
07-30-2020, 02:13 PM
Post: #6
RE: Estimation quiz!
(07-30-2020 10:54 AM)EdS2 Wrote:  which is larger, the number of millimetres in a mile or the number of square yards in a square kilometre?

We can estimate relative size without multiplying

1 yard = (36 in) (25.4 mm/in) = 914.4 mm → 1 yard ≈ 1 m

1 mile = 1760 yard ≈ 1.760e6 mm              // over-estimated 10%
1 km^2 = (1e3 m)^2 ≈ 1e6 yard^2           // under-estimated 20%

Since 1.76 > 1.30, we have mile/mm > km^2/yard^2

(07-30-2020 11:27 AM)Paul Dale Wrote:  Which is larger \( e^{\pi} \) or \( \pi ^ e \)?

e^x = 1 + x + x^2/2! + x^3/3! + ... ≥ 1+x
x ≥ ln(1+x)

→ if x≠0, x > ln(1+x)

ln(e^y) = y
ln(y^e) = e*(1 + ln(1 + (y/e-1))) ≤ e*(1 + (y/e-1)) = y

→ if y≠e, ln(y^e) < y ⇒ e^y > y^e
→ e^pi > pi^e
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Messages In This Thread
Estimation quiz! - EdS2 - 07-30-2020, 10:54 AM
RE: Estimation quiz! - Paul Dale - 07-30-2020, 11:27 AM
RE: Estimation quiz! - cruff - 07-30-2020, 01:00 PM
RE: Estimation quiz! - Paul Dale - 07-30-2020, 01:04 PM
RE: Estimation quiz! - cruff - 07-31-2020, 12:01 AM
RE: Estimation quiz! - KeithB - 07-30-2020, 01:56 PM
RE: Estimation quiz! - Paul Dale - 07-30-2020, 10:25 PM
RE: Estimation quiz! - Albert Chan - 07-30-2020 02:13 PM
RE: Estimation quiz! - Werner - 07-30-2020, 02:24 PM
RE: Estimation quiz! - Massimo Gnerucci - 07-30-2020, 02:34 PM
RE: Estimation quiz! - Albert Chan - 07-30-2020, 03:10 PM
RE: Estimation quiz! - rprosperi - 07-30-2020, 04:20 PM
RE: Estimation quiz! - Werner - 07-30-2020, 05:37 PM
RE: Estimation quiz! - Albert Chan - 07-30-2020, 06:10 PM
RE: Estimation quiz! - rprosperi - 07-30-2020, 08:03 PM
RE: Estimation quiz! - EdS2 - 07-31-2020, 01:26 PM
RE: Estimation quiz! - Paul Dale - 07-31-2020, 01:40 PM
RE: Estimation quiz! - Valentin Albillo - 07-31-2020, 02:33 PM
RE: Estimation quiz! - Paul Dale - 08-01-2020, 01:09 AM
RE: Estimation quiz! - EdS2 - 08-01-2020, 06:48 AM
RE: Estimation quiz! - Thomas Okken - 08-01-2020, 07:30 AM
RE: Estimation quiz! - Massimo Gnerucci - 08-01-2020, 10:09 AM
RE: Estimation quiz! - Valentin Albillo - 08-01-2020, 11:16 PM
RE: Estimation quiz! - ijabbott - 08-01-2020, 05:24 PM
RE: Estimation quiz! - johanw - 08-01-2020, 08:41 PM
RE: Estimation quiz! - J-F Garnier - 08-02-2020, 07:56 AM
RE: Estimation quiz! - SlideRule - 08-02-2020, 12:23 PM
RE: Estimation quiz! - Paul Dale - 08-02-2020, 01:10 PM
RE: Estimation quiz! - EdS2 - 08-02-2020, 03:13 PM
RE: Estimation quiz! - Valentin Albillo - 08-02-2020, 04:02 PM
RE: Estimation quiz! - Maximilian Hohmann - 08-02-2020, 05:04 PM
RE: Estimation quiz! - Valentin Albillo - 08-02-2020, 05:54 PM
RE: Estimation quiz! - Maximilian Hohmann - 08-02-2020, 09:01 PM
RE: Estimation quiz! - Albert Chan - 08-02-2020, 11:36 PM
RE: Estimation quiz! - SlideRule - 08-02-2020, 11:04 PM
RE: Estimation quiz! - Albert Chan - 08-02-2020, 05:41 PM
RE: Estimation quiz! - EdS2 - 08-03-2020, 07:56 AM
RE: Estimation quiz! - Albert Chan - 08-03-2020, 11:00 AM
RE: Estimation quiz! - pinkman - 08-03-2020, 03:08 PM
RE: Estimation quiz! - SlideRule - 08-03-2020, 11:57 AM
RE: Estimation quiz! - Valentin Albillo - 08-03-2020, 03:10 PM
RE: Estimation quiz! - KeithB - 08-03-2020, 01:54 PM
RE: Estimation quiz! - EdS2 - 08-03-2020, 03:22 PM
RE: Estimation quiz! - SlideRule - 08-03-2020, 03:49 PM
RE: Estimation quiz! - EdS2 - 08-04-2020, 10:17 AM
RE: Estimation quiz! - Thomas Okken - 08-04-2020, 12:33 PM
RE: Estimation quiz! - EdS2 - 08-07-2020, 06:57 AM
RE: Estimation quiz! - Thomas Okken - 08-07-2020, 09:03 AM
RE: Estimation quiz! - Albert Chan - 08-07-2020, 12:56 PM
RE: Estimation quiz! - EdS2 - 08-08-2020, 07:35 AM
RE: Estimation quiz! - Albert Chan - 08-08-2020, 05:01 PM
RE: Estimation quiz! - Csaba Tizedes - 08-04-2020, 11:25 AM
RE: Estimation quiz! - Csaba Tizedes - 08-07-2020, 02:06 PM
RE: Estimation quiz! - Albert Chan - 08-05-2020, 02:29 PM
RE: Estimation quiz! - KeithB - 08-05-2020, 03:38 PM
RE: Estimation quiz! - Albert Chan - 08-05-2020, 04:22 PM
RE: Estimation quiz! - StephenG1CMZ - 08-05-2020, 06:12 PM
RE: Estimation quiz! - EdS2 - 08-08-2020, 05:25 PM
RE: Estimation quiz! - KeithB - 08-13-2020, 01:41 PM
RE: Estimation quiz! - Albert Chan - 08-16-2020, 11:24 AM



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