Question

Design a sine function with the given properties. It has a period of 12 hr with a minimum value of -4 at \(t=0 \mathrm{hr}\) and a maximum value of 4 at \(t=6 \mathrm{hr}\)

Short answer

Question: Design a sine function to model a scenario where the minimum value is -4 at t=0 hours, the maximum value is 4 at t=6 hours, and the period is 12 hours. Answer: The sine function that models the given scenario is \(y(t) = 4\sin\left(\frac{\pi}{6}(t - 6)\right)\).

Step-by-step solution

Find the amplitude

The amplitude \(A\) is the distance between the maximum value and the equilibrium line or half of the distance between maximum and minimum values. Since the minimum value is -4 and the maximum value is 4, we can calculate the amplitude: \(A = \frac{4 - (-4)}{2} = \frac{8}{2} = 4\)

Find the period

Given that the period \(P\) is 12 hours, the sine function will complete one cycle in 12 hours.

Calculate the frequency

The frequency can be calculated using the period \(P\). Since the period is 12 hours, we have: \(f = \frac{1}{P} = \frac{1}{12\ \mathrm{hr}}\)

Calculate the phase shift

Maximum value occurs at \(t = 6 \mathrm{hr}\). Since the sine function should have its maximum value at 6 hours, that means the phase shift \(\phi\) would be equal to \(6\ \mathrm{hr}\) to shift the maximum value 6 hours forward: \(\phi = 6\ \mathrm{hr}\)

Write the sine function

Now we can plug in the amplitude, period, and phase shift into the standard form of the sine function: \(y(t) = 4\sin\left(\frac{2\pi}{12}(t - 6)\right)\) Thus, the sine function that meets the given properties is: \(y(t) = 4\sin\left(\frac{\pi}{6}(t - 6)\right)\)

Key concepts

Amplitude

When dealing with sine functions, understanding the amplitude is key. The amplitude of a sine function is how "tall" the waves are, or the maximum distance they travel from their equilibrium position. In simpler terms, it's half the distance between the maximum and minimum values of the function.
In our example, the maximum value is 4 and the minimum value is -4. To find the amplitude, we calculate the distance between these two points and then divide by 2. This is because the wave reaches 4 above and 4 below the central line, giving us an amplitude of 4. Here's the calculation:

• Distance between maximum and minimum: 4 - (-4) = 8
• Amplitude: \( \frac{8}{2} = 4 \)

The amplitude tells us how much the sine wave stretches away from its average height, which in other scenarios, such as sound waves, could relate to volume or intensity.

Period

The period of a sine function is the length of time it takes for the function to complete one full cycle. This means the wave repeats its pattern every certain number of units, like hours in this case.
In the given problem, the period is 12 hours. This indicates that the function starts at a certain point and returns to that same point after 12 hours of progression. In mathematical terms, the period \( P \) is expressed in the equation as how frequently the function repeats:

• Period \( P = 12 \) hours

Knowing the period is crucial, especially when modeling real-world phenomena like tides or sound waves, which exhibit repetitive cycles over time.

Phase Shift

Phase shift in a sine function moves the wave along the x-axis. It doesn't change the wave's shape or size, just its starting point.
For our sine function, the maximum value appears at 6 hours instead of 0, meaning there's been a horizontal shift in the sine wave. This shift is known as the phase shift. To compute it, understand where the wave's key points (like maxima or minima) occur:

• The function's maximum was found at \( t = 6 \) hours, clearly indicating a phase shift of 6 hours.
• In the equation, this phase shift is included as \( (t - 6) \).

Recognizing phase shifts is particularly important in contexts like electronics or signal processing, where the timing of wave phases can affect outcomes significantly.