A variable star is one whose brightness alternately increases and decreases. Lists of variable stars can

be found online, and many occur in familiar constellations visible with the naked eye. One such star, R
Hydrae, has an average magnitude rating of 7.2 but its magnitude varies by ±3.7 between its maximum
and minimum brightness. The period between two consecutive instances of maximum brightness (the
period) for R Hydrae is 389 Earth days. Design a function M(t) that models the brightness of R Hydrae
as a function of time.

To design a function that models the brightness of R Hydrae as a function of time, we can use a sinusoidal function since the brightness of a variable star typically follows a cyclic pattern.

Here's how you can design the function M(t):

1. Determine the amplitude: The magnitude varies by ±3.7 between its maximum and minimum brightness. The amplitude of a sine function is half of this range, so the amplitude would be (3.7/2) = 1.85.

2. Determine the period: The period between two consecutive instances of maximum brightness for R Hydrae is 389 Earth days. The period of a sinusoidal function is the length of one complete cycle. In this case, the period would be 389.

3. Determine the vertical shift: The average magnitude rating of R Hydrae is 7.2. The vertical shift of a sinusoidal function is the average value. In this case, the vertical shift would be 7.2.

4. Determine the phase shift: Since the problem does not mention any specific starting point, we can assume that the function starts at its maximum brightness. Therefore, there is no phase shift.

Putting it all together, the function M(t) can be expressed as:

M(t) = amplitude * sin((2 * π * t) / period) + vertical shift

Substituting the values we obtained:

M(t) = 1.85 * sin((2 * π * t) / 389) + 7.2

This function represents the brightness of R Hydrae as a function of time. You can use this function to calculate the brightness at any given time.