The average monthly temperature y in degrees Fahrenheit for the city of Fairbank

Question

The average monthly temperature y in degrees Fahrenheit for the city of Fairbanks, Alaska, fluctuates periodically between a low of -11.5 degree in January and a high of 61.3 degree in July. (a) Find a trigonometric function f(t) that models the average temperature of Fairbanks, where t is months after January. (b) Without solving the equation, how many solutions should the equation f(t) = 23 have for 0 lessthanorequalto t lessthanorequalto 12? (c) Solve the equation (with the domain restrictions) from part (b), and interpret your results in context. (d) In the southern hemisphere, the times of the year at which summer and winter occur are reversed relative to the northern hemisphere. Find a formula that models the average temperature of a city in the southern hemisphere whose average temperatures are similar to those of Fairbanks.

Explanation / Answer

a).An equation that can be used to model these data is of the form

     f(t)=AsinB(t+C)+D

         where A,B,C,D are constants

f(t) is the temparature in °C and t is the month (1–12).

A = amplitude = (tmax - tmin)/2
B = 2/12
C = units translated to the right
D = tmin + amplitude = units translated up

A=(tmax-tmin)/2

tmax- tmin= 61.3-(-11.5)=72.8

A=36.4

D=tmin +amplitude=-11.5+36.5=24.9

answer : y(t)=36.4sin((t+1)/6)+24.9

b).As the given equation is a sin function

    the general solution of y is 2n ± t

    ie , we will get two solutions for each

c) y = AsinB(t+C)+D

     36.4sin((t+1)/6)+24.9=23

     36.4sin(t+1)/6=-1.9/36.4

     (t+1)/6=-0.05219

      t+1=-5.7135

      t=-6.7135

d)

    y=AsinB(x+C)+D

or

   y=AcosB(x-C)+D

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