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 tr1sth3t
Lieutenant Commander
Joined: Feb 27, 2007
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Posts: 279
Location: Bergen
    
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 Posted:
Mon Mar 22, 2010 6:53 pm Post subject: brainTeaser[1] |
 
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The elongation (we'll say α) of a planet is the angle formed by the planet, Earth and the Sun. It is known that the distance from the Sun to Venus is 0.723AU and from the Earth to the Sun is 1AU. At a certain time the elongation of Venus is found to be 39.4°. Find the possible distances from the Earth to Venus at that time in Astronomical Units (AU).
Hint: Use the law of sines.
sin(A)/a = sin(B)/b = sin(C)/c
ANSWER TO BRAIN TEASER 0:
| Quote: |
r = 10m, Δr = ±1.2m
h = 15m, Δh = ±2.4m
If we logically think about these changes (Δr and Δh) in r and h they are simple the limits on the instrument.
ΔV ≈ dV = (dπr²)
= πr²dh + hd(πr²)
= πr²dh + h2πrdr
= πr²dh + 2πrhdr
|ΔV| ≈ |dV| = |πr²dh + 2πrhdr|
≤ |πr²dh| + |2πrhdr|
= |π(10)²(±2.4)| + |2π(10)(15)(±1.2m)|
= 240π + 360π = 600πm³ ≈ 1,884.66m³
Now we need to compute the relative error.
V = πr²h = (10)²(15)π = 1500π ≈ 4,712.4m³
|ΔV ÷ V| ≈ |dV ÷ V| = 600πm³ ÷ 1500πm³
Here the units cancel along with the symbol π (pi) leaving:
600 ÷ 1500 = 0.40 or a 40% relative error.
(a) The maximum possible absolute error in the calculated volume is 1,884.66m³
(b) The maximum possible relative (percentage) error in the calculated volume is 40%.
So this instrument sucks and a new one needs to be bought. |
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Last edited by tr1sth3t on Sun Apr 04, 2010 3:43 am; edited 5 times in total |
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 irule
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Joined: Sep 20, 2006
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Posts: 1379
Location: Canada
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Mon Mar 22, 2010 10:34 pm Post subject: |
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Yay trig. I'm actually okay at this. I got a minimum distance of 0.02013 au and a max of 1.15325 au. And I'm probably completely wrong.
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tr1sth3t
Lieutenant Commander
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Posts: 279
Location: Bergen
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Mon Mar 22, 2010 10:58 pm Post subject: |
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| irule wrote: |
| Yay trig. I'm actually okay at this. I got a minimum distance of 0.02013 au and a max of 1.15325 au. And I'm probably completely wrong. |
I will say that the correct answer fall in between your max and min.
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It is more likely that a brain randomly forms out of the chaos with false memories of its life than that the universe around us would have billions of self-aware brains. |
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irule
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Mon Mar 22, 2010 10:59 pm Post subject: |
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Isn't the possible distance any value between those two?
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tr1sth3t
Lieutenant Commander
Joined: Feb 27, 2007
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Posts: 279
Location: Bergen
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Posted:
Mon Mar 22, 2010 11:52 pm Post subject: |
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| irule wrote: |
| Isn't the possible distance any value between those two? |
Yes, but which value is it? Specifically the one where the angle formed by the Earth is 39.4°.
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It is more likely that a brain randomly forms out of the chaos with false memories of its life than that the universe around us would have billions of self-aware brains.
Last edited by tr1sth3t on Sun Apr 04, 2010 3:37 am; edited 2 times in total |
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irule
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Tue Mar 23, 2010 12:07 am Post subject: |
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I thought the 39.4 degree angle was formed by Venus?
Nvm... the key lies in the fact that Earth is farther away from the Sun than Venus is... so it is between 1.1173728 and 26.849632? I don't see how there is enough information to get a single value. |
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tr1sth3t
Lieutenant Commander
Joined: Feb 27, 2007
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Posts: 279
Location: Bergen
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Posted:
Sun Apr 04, 2010 4:04 am Post subject: |
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ANSWER:
V1 and V2 are the positions of Venus, E is the earth and S is the Sun. Since the distance from the Sun to Venus are the same consider SV1 = SV2 = 0.723AU. Consequently, the corresponding opposite angles are equal. So, the remaining angles are measured.
Now, using law of sines we have
[ V2E ÷ sin(xº) ] = [ 1 ÷ sin(140.6 - x)º ] = [ 0.723 ÷ sin(39.4º) ]
⇒ V2E = sin(xº)[ 0.723 ÷ sin(39.4º) ]
also, sin(140.6 - x)º = [ sin(39.4º) ÷ 0.723 ]
⇒ xº = 140.6º - sinˉ¹[ sin(39.4º) ÷ 0.723 ] = 79.208º
∴ V2E = 1.11891 astronomical units.
Again from the triangle V1SV2 , using the sine law we get
[ 0.723 ÷ sin(140.6 - x)º ] = [ V1V2 ÷ sin(101 - 2x)º ]
⇒ V1V2 = -0.708079 AU
∴ V1E = 1.11891-0.708079 = 0.42 AU
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It is more likely that a brain randomly forms out of the chaos with false memories of its life than that the universe around us would have billions of self-aware brains. |
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 DarkVampire
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Joined: Apr 15, 2010
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Location: Friesland
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Posted:
Sun May 22, 2011 3:47 pm Post subject: |
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hmm i dont know the anwser
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