i really do not understand

Question

Over a 24-hour period, the temperature in a town can be modeled by one period of a sinusoidal function. The temperature measures 70°F in the morning, rises to a high of 80°F, falls to a low of 60°F, and then rises to 70°F by the next morning.

What is the equation for the sine function f(x), where x represents time in hours since the beginning of the 24-hour period, that models the situation?

Answer 1

Start with an initial temperature of T₀ = 70° at t = 0 hours, and add to that a sine function that does the following, in order:

 

Rises by 10°, reaching 80° after 6 hours

Drops by 10º, reaching 70° after 12 hours

Drops by 10°, reaching 60° after 18 hours

Rises by 10°, reaching  70° after 24 hours

 

Note that the amplitude of the sine function is A = 10°.

 

T = period of the sine function

T = 24 hours

 

f(x) = T₀ + A sin[(2π/T)x]

 

where

 

T₀ = 70°

A = 10°

T = 24 hours

x = elapsed time in hours

 

f(x) = 70º + (10°)sin[(π/12)x]

 

with x in hours.

 

Test this solution and see that it fits the data.  (Make sure your calculator is in the radian mode!)