Question

Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 100 degrees and the low temperature of 70 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. Assume the next low is 24 hours later.

Answer 1

Answer:

The function for the outside temperature is represented by $T(t) = 85\º + 15\º \cdot \sin \left[\frac{t-6\,h}{24\,h} \right]$, where t is measured in hours.

Step-by-step explanation:

Since outside temperature can be modelled as a sinusoidal function, the period is of 24 hours and amplitude of temperature and average temperature are, respectively:

Amplitude

$A = \frac{100\º-70\º}{2}$

$A = 15\º$

Mean temperature

$\bar T = \frac{70\º+100\º}{2}$

$\bar T = 85\º$

Given that average temperature occurs six hours after the lowest temperature is registered. The temperature function is expressed as:

$T(t) = \bar T + A \cdot \sin \left[2\pi\cdot\frac{t-6\,h}{\tau} \right]$

Where:

$\bar T$ - Mean temperature, measured in degrees.

$A$ - Amplitude, measured in degrees.

$\tau$ - Daily period, measured in hours.

$t$ - Time, measured in hours. (where t = 0 corresponds with 5 AM).

If $\bar T = 85\º$, $A = 15\º$ and $\tau = 24\,h$, the resulting function for the outside temperature is:

$T(t) = 85\º + 15\º \cdot \sin \left[\frac{t-6\,h}{24\,h} \right]$