The number of hours of daylight in Winnipeg, Manitoba, can be approximated by d(

Question

The number of hours of daylight in Winnipeg, Manitoba, can be approximated by d(t) = 4.15s in(0.0172 t - 1.377) + 12 where t is measured in days where t = 1 corresponds to January 1st. t = 2 corresponds to January 2nd. etc. a. Evaluate d(15) and interpret your answer in terms of this problem. b. Find the first day of the year that has 11 hours of sunlight. c. Find the period of d(t) and breifly explain why this period is expected. [Recall that for this course, angles are always measured in radians]

Explanation / Answer

d = 4.15sin(0.0172t - 1.377) + 12

d(15) :
4.15sin(0.0172*15 -1.377) + 12
= 8.26639311452505

The number of hrs of daylight in Winnipeg
on jan 15th is approximately 8.27 hrs

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b)
First day that has 11 hrs sunlight :

11 = 4.15sin(0.0172t - 1.377) + 12

sin(0.0172t - 1.377) = -1/4.15

0.0172t - 1.377 = 3.384 + 2n*pi , 6.04 + 2n*pi

Solving for t here, we get :
t = (4.761 + 2n*pi)/0.0172
and t = (7.417 + 2n*pi)/0.0172

We need to plug in random integers n and find the first day

We need to plug in n = -1 into the second solution there...
65.919

This means, the first day with 11 hrs sunlight is
the 66th day, i.e march 7th

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c)
PEriod :

The number multiplied to t is : 0.0172

Period = 2pi/0.0172

Period = 365.3014713476503766

as in approx 365, which is expected
as we have 365 days in a year

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