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January 1st, 2009, 05:17 PM  #1 
Newbie Joined: Dec 2008 Posts: 3 Thanks: 0  TRIG:On what day(s) of the yr we have exctly 12h of daylight
Hi. I am a grade 12 student and I was wondering if someone can help me with this question. The length of daylight cycles periodically approx. every 365 days. If one takes January 1st as day 1, then day 172 (June 21) is the longest day of the year with 15.5 h of daylight. Day 355 (December 21) is the shortest day of the yr with about 8.5 h of daylight. Using an algebraic model (ex. equation you come up with) to calculate on what day(s) of the year we have exactly 12 h of daylight. I know it is a cosine function with amplitude of 12h and period of 365 thus k = (2?)/365. This is what i have so far 3.5cos[(?/173)(x+173)]+12 when you solve.. x = 86, 268 but then one of my friend got 3.5cos(72/73 (x+10))+12 and when you solve.. x = 81.25, 263.75 and my other friend got 7cos(730?t172)+12 << 172 because maximum is at 172 ?? I'm not sure which one is right. or all of those may be wrong.. does anyone know how to solve this question? It will be great if anyone can help with this question, Thank you. Also... 2) The population size, P, of owls (predatory) is modeled by function P (t) = 2000 + 500sin0.25(t?/12), where t rep time in months. The pop size, p, of mice(prey) is given by p(t)= 10000 + 2000cos(t+?/12). Determine the time, to nearest tenth of a month, during the first two years when the pop of the mice is exactly 5 time the pop of the owls. Note that theses equation are both in radians measure. So I've started with..p(t)= 5P(t) 10000 + 2000cos(t+?/12) = 5 [2000 + 500sin0.25(t?/12)] 10000 + 2000cos(t+?/12) = 10000+ 1000sin0.25(t?/12) 2000cos(t+?/12) = 1000sin0.25(t?/12) then divide both by 2000 cos(t+?/12) = 0.5sin0.25(t?/12) cos(t+?/12)  0.5sin0.25(t?/12) = 0 then what do i do afterwards?? I attempted to change cos(t+?/12) to sin using double angle formula.. cos(t+?/12) => cost cos?/12  sint sin?/12 but doesn't seem to work.. or should i approach it in a different way ? Again, it'd be great if someone can help me out. Thank you in advance. :] 
January 1st, 2009, 05:32 PM  #2 
Newbie Joined: Jan 2009 Posts: 14 Thanks: 0  Re: TRIG:On what day(s) of the yr we have exctly 12h of daylight
I've only finished Gr.9 math so I don't know how to help you with your method for the first question, but by realizing 12 is between 9.5 and 15.5 the answer would just be the days exactly between 355 and 172, which through simple math can be figured out that they are 263(.5) and 81(it gets a bit messed up as there aren't an even number of days in a year). Maybe knowing the answer will help you figure out the actual way to solve it.

January 2nd, 2009, 01:01 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,581 Thanks: 610  Re: TRIG:On what day(s) of the yr we have exctly 12h of daylight
"what day(s) of the year we have exactly 12 h of daylight" In reality, these are supposed to be the first day of spring and first day of fall. The actual dates take into account the fact the earth's orbit is not exactly circular. 

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12h, daylight, days, exctly, trigon 
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