Outside temperature over a day can be modeled as....

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 96 degrees occurs at 5 PM and the average temperature for the day is 85 degrees. Find the temperature, to the nearest degree, at 9 AM.

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Answer 1

Hi Cynthia G.,

If the high temperature is 96^o and the average temperature is 85^o (difference of 11^o), then the low temp. would be 74^o (85^o - 11^o).

For a sinusoidal wave [y = A*sin(B*(x - C)) + D] the amplitude would be A = 11.

For a 24 hour day the period is 24 and B = 2π/period = 2π/24 = π/12.

Using a 24 hour time period, 5pm is the 17^th hour. Lets choose to model with a cosine wave which will have maximums at 0^th hour, 24^th hour, .... We need a maximum at the 17^th hour, so we need to shift the graph 7 hours to the left (move the 24^th hour maximum to make it a 17^th hour a maximum), so C = -7.

We will also need to shift the graph up to the average, D = 85.

The final equation becomes y = 11*cos(π/12*(t + 7)) + 85, or, y = 11*cos(π*t/12 + 7π/12) + 85.

For 9am we can set t = 9 to get y = 79.5^o.

I hope this helps, Joe.