Modeling with Trig Functions (Section 6.5)

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 52 and 68 degrees during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight, to two decimal places, does the temperature first reach 58 degrees?

Hours:

Answer 1

Average temperature = (52+68)/2 = 60

Range is 52-68

Asin(B(x+C)) + D = y(x) range of sine is [-1,1]

Period = 2π/B = 24 hours --> B = π/12

A + D = 68

-A+D = 52

2D = 120 --> D = 60, A= 8

when x = 10, y(10) = 60

8sin((π/12)(10+C)) + 60 = 60 --> sin(θ) = 0 --> θ = 0

Thus (π/12)(10+C) = 0 ---> C = -10

y(x) = 58 = 8sin((π/2)(x-10)) + 60 --> sin((π/2)(x-10)) = -1/4 --> sin^-1(-0.25) = 0.2527 = (π/2)(x-10)

x = 9.03 hours after midnight