Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 75 degrees and the low temperature of 35 degrees occurs at 4 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer: D (t) = 20°*cos ((pi/4) * t) + 55°

Step-by-step explanation:

using that t = 0 is midnight, we know.

We know:

Max temp = 75°

Min temp = 35° (occurs at t = 4 hours)

Now we can model this as:

D (t) = A*cos (c*t) + B

Where A, c and B are constants.

we have a minimum at t = 4 hours, a minimum means that cos (c*t) = - 1

then we have that:

D (4) = A*cos (c*4) + B = A * (-1) + B = 35°

here we also have that cos (c*4) = - 1

this means that c*4 = pi

c = pi/4

We also have that the maximum temperature is 75°, the maximum temperature is when cos (c*t) = 1

D (t0) = A * (1) + B = 75°

with this we can find the values of A and B.

-A + B = 35°

A + B = 75°

We isolate B in the first equation and then replace it in the second equation.

B = 35° + A

A + B = 75°

A + 35° + A = 75°

2A = 40°

A = 40°/2 = 20°

B = 35° + 20° = 55°.

Our equation is:

D (t) = 20°*cos ((pi/4) * t) + 55°