Calculus (6th Edition) Chapter 1.4, Problem 14E The table shows the normal daily

Question

Calculus (6th Edition)

Chapter 1.4, Problem 14E

The table shows the normal daily high temperatures for Miami M and Syracuse S (in degrees Fahrenheit) for month t, with t = 1 corresponding to January.†

(a) The following is a model for Miami.

M(t) = 83.70 + 7.46 sin(0.4912t 1.95)

Find a model of the form S(t) = D + Asin(t + ) for Syracuse. (Round your coefficients to three decimal places.)

(b)Use the models to estimate the average annual temperature in each city.

(c)What is the period of each model? (Round your answer to three decimal places.)

t 1 2 3 4 5 6 7 8 9 10 11 12 M 76.5 77.7 80.7 83.8 87.2 89.5 90.9 90.6 89.0 85.4 81.2 77.5 S 31.4 33.5 43.1 55.7 68.5 77.0 81.7 79.6 71.4 59.8 47.4 36.3

Explanation / Answer

a > S(t) = D + Asin(t + )

D=average temperature of Syracuse
D = [31.4+33.5+43.1+55.7+68.5+77.0+81.7+79.6+71.4+59.8+47.4+36.3]/12

D = 57.12

A = [Max. temperature - Min. temperature]/2= [81.7 - 31.4]/2 = 25.15

The period is 12 months.

=> period =12 = 2/

=> . = /6= 0.524

= 1.95 is the phase shift. It should be the same for both. so for Syracuse:

S(t) = 57.12 + 25.15sin(0.524t + 1.95)

b> average temperature of M is = 83.7 degree F (This is the avarage of the teperatures for 12 months for Miami )

average temperature S : = 57.12 degree F

c> for Syracuse The period is 2/ = 2 / /6 = 12 months

for Miami the period is= 2pi/0.4912 = 12.792 months

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