Outside temperature over a day can be modeled as a sinusoidal function

Question

Suppose you know the temperature varies between 45 and 55 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 49 degrees?

Answer 1

45.67 degrees at 2 hours after midnight

50 at 10 am, 50 at 10pm, 45 at 4am, 55 at 4pm

at 2am the temperature is between 45 and 50, closer to 45 though

2am is 2/3 of the way from 10pm to 4am

2/3 of .866 (-sin240) is .67. 45 + .67 = 45.67 degrees

average temperature is (45+55)/2 = 50 degrees, occurring at 10 am

mid line is 50, max 55, min 45. f(x) = 5sin(bx+c) + 50

general equation is asin(b+c)+d where d= the mid line, a = half of (max - min). b concerns frequency or period. b = 360/period = 360/24 = 15. the period is 24 hours. Temperatures repeat every 24 hours. c is the phase shift.

period for sinx is 2pi = 360

period for temperature change is 24 hours

f(x) = 5sin(15x -150) + 50, set = 49 and solve for x

measure x similar to military time, or x = 10 + the number of hours after 10am

x = 22.77 = 10.77 pm which is not "after midnight'

or, 2nd solution is

x = 33.23 = 9:23 am, the 2nd time the temperature is 49 degrees

if the original 10AM is Monday morning, then 10.77PM is Monday night and 9:23AM is Tuesday morning, after midnight.

Graphically it's a sine curve where 10AM is 50 degrees, 10PM is 50 degrees and 10AM the next day is also 50 degrees. 49 degrees occurs just before 10AM the next day, or 9:23AM

Try graphing it It helps