Outside tempature

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature for the day is 68 degrees and the low temperature of 32 degrees occurs at 5 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer 1

The general form of the sinusoidal function is

D(t) = M + Asin(2πt/T + p)

where

M is the mean temperature,

A is the amplitude,

T is the period,

p is the phase.

The mean

M = (68+32)/2 = 50

The amplitude is the maximum deviation from the mean

A = 68-50 = 18

The period

T = 24 hours

The phase is calculated from the requirement that the lowest temperature occurs at 5am.

The sine function reaches its minimum when its argument is 3π/2.

2πt/T + p = 3π/2

2×π×5/24 + p = 3π/2

p = 13π/12

So the whole equation is

D(t) = 50 + 18sin((t+13)π/12)