Weather Modeling Activity

Question

Based on the last 5 years, the hottest day of the year in Michigan is generally around August 10th (the 222nd day of the year, assuming the year has 365 days), which reaches an average temperature of 95℉. The coldest day of the year is typically around −8℉. Assuming that the temperature is periodic from year to year, model the temperature with an equation in the form 𝑇(𝑥) = 𝑎 cos(𝑘(𝑥 − 𝑏)) + 𝑑 where 𝑥 is the number of days as follows:

a) What would be the period 𝑝?

b) Find the appropriate value of 𝑘, using the period 𝑝.

c) What would be the phase shift (horizontal shift) 𝑏? (Hint: cos 𝑥 has its highest value when 𝑥 = 0. We want to shift so that the highest value occurs during the hottest day.)

d) Find the amplitude 𝑎.

e) Find 𝑑 (Hint: Use the amplitude and the maximum or minimum temperature.)

f) Write the equation for this model.

g) The weather forecast for October 31st (the 304 th day of the year) was predicted to be 51℉. How does this compare to the value predicted by the model?

h) Find 𝑥 as a function in terms of 𝑇. Determine the range of the function 𝑥(𝑇).

i) According to this model, when would be the first time during the year that the temperature reaches 75℉?

Answer 1

a) period p = 365

b) k = 2π/365

c) k(x-b) = 0 --> b = 222

d) a+d = 95

-a+d = -8

Adding the two equations, d = 43.5, a = 51.5

e) d = 43.5 from answer d)

f) T(x) = 51.5cos((2π/365)(x-222)) +43.5

g) T(304) = 51.6° pretty close to 51°

h) x(T) = arccos(((T-43.5)/51.5)(365/2π) + 222

i) T(75) = 275 --> October 2nd