In Philadelphia the number of hours of daylight on day t whe

In Philadelphia the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function L(t) = 12 + 2.83 sin(2365(t 80)) a) Which days of the year have about 10 hours of daylight? b) How many days of the year have more than 10 hours of daylight?

Solution

If the function is L(t) = 12 + 2.83 sin(2365(t 80)), then

a) The days of the year have about 10 hours of daylight

10 = 12 + 2.83 sin(2365(t 80))

(sin ^{-1} (10 -12 / 2.83))/730 = (t 80))

Thus t = 80 + (sin ^{-1} (-2) / 2.83)) / 730

or      t = 80   

                           .... Since -1<= (sin ^{-1} (-2) / 2.83)) / 730 <=1         

Hence for t = 80 days or march 21.

b. 365-80 = 285 days                                                  Ans

One is March 21 and other day is after 285 days of winter which is oct 13.

There are 2 days in a year.

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If the function is L(t) = 12 + 2.83 sin(2/365(t 80)), then

10 = 12 + 2.83 sin[(2/365*(t-365)]

10-12 = 2.83 sin[(2*/365)(t-365)]

-2 / 2.83 = sin[(2*/365)(t-365)]

-0.7067 = sin[(2*/365)(t-365)]
[(2/365)(t-365)] = sin^-1 ( -0.7848)

or   [(2*pi*/365)(t-365)] = -0.90244

Thus

        t - 365 = -52
or    t = -52+365        

      t = 313 days
Therefore on Oct 11 about 10 hours of daylight.

b. One day is on Oct 11 and the other day is Feb 11 (52 days after winter), which are about 10 hours of daylight.

    Hence there are 2 days in a year.