The number of hours in daylight in a city is given by S(x) = 3.15sin[0.0172(x-83
ID: 2883072 • Letter: T
Question
The number of hours in daylight in a city is given by S(x) = 3.15sin[0.0172(x-83)] + 12.15, where x is the xth day of the year (Jan 1=1, Jan 2=2, Dec 31=365)
a. find the maximum number of hours of daylight. On what day of the year does this occur?
b. find the minimum number of hours of daylight. on what day of the year does this occur?
c. what is the average number of hours of daylight? (don't do complicated calc.. use the symmetry of a graph) explain answer.
d. what is the period of this function? what is the meaning of that number?
e. is this city in the Northern Hemisphere or Southern Hemisphere? Explain please.
Explanation / Answer
a) max :
This would happen at sin(pi/2)...
So, 0.0172(x - 83) = pi/2
x - 83 = 91.3253678369125941
x = 174.3253678369125941
So, approx x = 174
And this is on May 23rd
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b) Min :
This would happen at sin(3pi/2)
0.0172(x - 83) =3pi/2
x = 356.9761035107377824
x = 357 approx
This is approx DEc 23rd
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c)
Average = (max + min)/2
Max = 3.15sin(pi/2) + 12.15 = 15.3 hrs
Min = 3.15sin(3pi/2) + 12.15 = 9 hrs
Average = (15.3 + 9)/2 = 12.15 hrs
So, an avg of 12.15 hrs of sunlight
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d)
Period = 2pi / b
Period = 2pi / 0.0125
Period = 365.3014713476503766
So, period is approx 365
which is the number of days to the year of course
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e)
Northern hemisphere