Question

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 1

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

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°