Question

Design a sine function with the given properties. It has a period of 24 hr with a minimum value of 10 at \(t=3\) hr and a maximum value of 16 at \(t=15 \mathrm{hr}\)

Short answer

Answer: The sine function that models the given properties is $y(t) = 3\sin\left(\dfrac{2\pi}{24}(t-\dfrac{3\pi}{2})\right) + 13$.

Step-by-step solution

Determine the amplitude

The amplitude is half the difference between the maximum and minimum values. So, in this case, it's \((16-10)/2=3\). The amplitude is 3.

Determine the period

The exercise states that the period of the sine function is 24 hours.

Determine the frequency

Now we need to find the frequency, which is the reciprocal of the period. The frequency is \(\dfrac{1}{24}\).

Determine the phase shift

Since the minimum value occurs at \(t=3\) hr, we have to find the value that shifted the sine function horizontally. The sine function usually has its minimum value at \(\dfrac{3\pi}{2}\), so the phase shift is \(3 \mathrm{hr} - \dfrac{3\pi}{2 \cdot \dfrac{1}{24}} = \dfrac{3\pi}{2}\).

Determine the vertical shift

The vertical shift is the average of the minimum and maximum values. In this case, it's \((10+16)/2=13\). The vertical shift is 13.

Combine the values to create the sine function

The general form of the sine function with amplitude, period, phase shift, and vertical shift is given by: $$ y(x) = A \sin(B(x - C)) + D $$ where \(A\) is the amplitude, \(\dfrac{2\pi}{B}\) is the period, \(C\) is the phase shift, and \(D\) is the vertical shift. Plugging in the values, we found earlier, we get: $$ y(t) = 3\sin\left(\dfrac{2\pi}{24}(t-\dfrac{3\pi}{2})\right) + 13 $$ This is the desired sine function with the given properties.

Key concepts

Amplitude

The amplitude of a sine function is a measure of how far the function's graph stretches vertically from its central axis, which is usually the horizontal line around which the sine wave oscillates. Imagine it as the height of the wave from the middle line to its peak or trough. In mathematical terms, for a sine function described by

• y = A sin(B(x - C)) + D,

the amplitude is represented by |A|. In the context of our function, the amplitude determines the distance between the minimum and maximum values of the wave.

To find the amplitude, we take half the difference between the maximum and minimum values of the function. In our example, where the maximum value is 16 and the minimum value is 10, we calculate the amplitude as:

• \[\text{Amplitude} = \frac{16 - 10}{2} = 3\]

This tells us that the function oscillates 3 units above and below its average value, which is vital for describing the height of the entire wave. Understanding amplitude helps in visualizing how intense or pronounced the wave is.

Period

The period of a sine function is the duration it takes for the function to complete a full cycle of its wave pattern. You can think of it as the time it takes for a wave to go from start to finish and then repeat. Understanding period is crucial because it describes the spacing between the repeating patterns or peaks in the sine wave.

The general form of the function

• y = A sin(B(x - C)) + D,

indicates how the period is calculated through the equation

• \[\text{Period} = \frac{2\pi}{B}\]

In our case, the problem defines a period of 24 hours. This means every 24 hours, the wave repeats its pattern. This repetition is key for functions that model cyclical events like daily temperatures or tides.

In practice, knowing the period allows us to predict how often the wave reaches peak values, which is helpful for understanding the rhythm or timing of naturally occurring processes.

Vertical Shift

The vertical shift of a sine function is the amount it moves up or down from the usual horizontal axis, typically the x-axis. This concept is essential for identifying where the central line of the wave is on the graph. The vertical shift effectively changes the base level of the wave, around which the oscillations happen. This is particularly useful when you want the sine wave to start at a different value than the standard zero.

For a sine function of the form

• y = A sin(B(x - C)) + D,

the vertical shift is represented by D. It can be thought of as the average of the maximum and minimum values of the function. In the given problem, the vertical shift is calculated as:

• \[\text{Vertical Shift} = \frac{10 + 16}{2} = 13\]

This vertical shift means that the "new" middle of the wave is at 13, setting the axis around which the wave oscillates. Knowing this helps in determining the range over which the wave moves and can be crucial when matching data to models.