Pre calculus question

Question

In a certain city 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)= 13 + 2.83 sin(2pi/365(t-80))

(a) Which days of the year have about 11 h of daylight? (Enter your answers as a comma-separated list.)

(b) How.many days of the year have more than 11 h of daylight?

Answer 1

L(t) = 13 + 2.83sin(2pi/365(t- 80))

general form of the sinoidal polynomial

is

Asin(B(t-C))+D,

A = amplitude = 2.83

B = 1/period = 2pi/365

period = 365/2pi = about 58.1 days= about 2 months

C= 80 = phase shift

D = 13 = midline = average time= midpoint between max and min

11 = 13+2.83sin(2pi/365(t-80))

-2/2.83 = sin2pi/365(t-80)

2pi/365(t-80) = sin^-1(-2/2.83) = about .707 = about sqr2/2

(pi/365)(t-80) = sqr2

t-80 = (sqr2)(365/pi)

t= 80 +pisqr2/365 = about 80.012

80 days after Jan. 1 = 31+28+21

= March 21st plus a tiny time period,

maybe early March 22= the spring equinox

about 12:53 am

= about 1am Mar. 22 is one day

add the cycle period to get more day(s)

2pi/365 = B

B=1/period = 365/2pi = about 58.1 days

3/22 + 58 = about May 19

add another 58 days

July 16,

add another 58

Sept 13

add 58

Nov. 10

add 58

Jan. 27 of the next year

3/22, 5/19, 7/16, 9/13, 11/10 for year one

but also add in the half period times

58.1(5)/2 + 51 = 196 1/4 days

vs. 365.25 - 196.25 = 169 days

of <11 and >11

no guarantees of a mistake(s) or two

but it's the general method

know how sine curves work

maybe use a graphical calculator to solve

or check your work