Find the answer in hours.

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 26 and 64 degrees during the day and the average daily temperature first occurs at 9 AM. How many hours after midnight, to two decimal places, does the temperature first reach 36 degrees?

Answer 1

f(t) = Asin(B(t-C)) +D

A = Amplitude = (max-min)/2 = 64-26)/2 = 32-13 = 19

B = frequency = 2pi/period =2pi/24 = pi/12, period =24 = 2pi/B

C = phase shift

D = midline = vertical shift = average of max&min = (64+26)/2 = 32+13 = 35

f(t) = 19sin(B(t-C)) + 35

35 at 9 am, sin(B(t-C) = 0 = sin(Bt -BC), Bt-BC = arcsin0 = 0, t=C=9

f(t) = 19sin(B(t-9)) +35

f(t) = 19sin(pi/12)(t-9)) + 35

reaches 36 degrees after midnight somewhere between t=15 and maybe 20 or so

36 -35 = 19sin(pi/12)(t-9))

sin(pi/12)(t-9)) = 1/19

(pi/12)(t-19) = arcsin(1/19) = about 0.0526559 radians

t-19 = (12/pi)(.0526559)

t = 19 +.216

t = 19.216

= 4.216 hours after midnight = about 4:13 am

35 at 9am, halfway between is 9pm with 64 degrees

9pm to 4:13am is about 7 1/4 hours, and from 4:13 am to 9am is about 4 3/4 hours until hitting 35 degrees