The outside temperature over the course of a day can be...

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is $68^{\circ} \mathrm{F}$ at midnight and the high and low temperatures during the day are $80^{\circ} \mathrm{F}$ and $56^{\circ} \mathrm{F}$ , respectively. Assuming $t$ is the number of hours since midnight, find a function for the temperature, $D,$ in terms of $t .$

Key Concepts

Sinusoidal Functions

Sinusoidal functions are periodic functions that oscillate in a smooth, continuous wave-like pattern, typically modeled using sine or cosine functions. They are widely used to represent repeating phenomena in various fields, such as temperature variations over a day, sound waves, and seasonal changes.

Amplitude

The amplitude of a sinusoidal function is the distance from the midline to the maximum (or minimum) value of the function. It represents half the range of the oscillation and indicates how much the function's values deviate from the average value.

Midline (Vertical Shift)

The midline is the horizontal line around which the sinusoidal function oscillates. It is determined by taking the average of the maximum and minimum values. The vertical shift moves the basic sinusoidal curve up or down to align with the midline of the given data.

Period

The period of a sinusoidal function is the length of the interval over which the function completes one full cycle of its pattern. In models involving time, such as daily temperature variations, the period is essential in capturing the cyclical nature of the changes.

Phase Shift

The phase shift in a sinusoidal function is the horizontal translation of the function's graph. It determines where the function starts its cycle relative to the origin of the coordinate system. Adjusting the phase shift aligns the model with real-world timing events, such as the time of day when the temperature reaches a particular value.

Transcript

00:01 All right, so here we know that the temperature in a certain city can be graphed as a sinusoidal graph.

00:09 And there are a couple of things that we know.

00:10 We know that the temperature is 68 degrees at midnight.

00:15 Midnight is going to be t equals zero.

00:17 It gets to the peak of 80 degrees.

00:23 And the low is 56 degrees.

00:27 And we want to know what d of t is.

00:32 This function, given that d is a function for the temperature in terms of t time.

00:39 All right.

00:40 So the first thing we can recognize is that this is actually not based on the x -axis at all.

00:44 It's shifted up to y -equal 68.

00:47 So a more accurate version of this graph would look something like this.

00:58 So we have, here's the x -axis.

01:01 Here's y -equal 68.

01:04 This will be 12 hours, and this will be 24 hours.

01:09 All right, so we know that it reaches a peak at 80 degrees fahrenheit, and it reaches a minimum at 56 degrees fahrenheit.

01:22 So we kind of have to figure out if the graph is going to look like this or if it's going to look like this.

01:30 And because warmer temperatures happen later in the day, i'm inclined to believe that the graph looks more like this, right? so at midnight at 68 degrees, it cools down to like 4 a .m.

01:42 6 a .m.

01:42 Is usually the coldest part of the day, so it heats up, usually around 4 p .m.

01:46 That's when the highest point temperature is, and then it cools back down to 60 degrees at midnight.

01:53 So a couple of things we can tell...

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