A variable star is one whose brightness alternately increases and decreases. For one such star, the time between periods of maximum brightness is 5.6 days, the average brightness (or magnitude) of the star is 5.5, and its brightness varies by ±0.30 magnitude. Find a function that models the brightness of the star as a function of time (in days), t. (Assume that at t = 0 the brightness of the star is 5.5 and that it is increasing.)
since we have our minimum at t=0, the function will look like
y = -cos(kt)
The brightness varies by ±0.30, so that is the amplitude.
y = -0.30 cos(kt)
Since the low is 5.5, the axis of the curve is at 5.5+0.30, so
y = 5.80 - 0.30cos(kt)
The period of cos(kt) is 2π/k, so we must have
2π/k = 5.6, or k = π/2.8
y = 5.80 - 0.30cos(π/2.8 t)
To model the brightness of the star as a function of time, we can use a sine function since it alternates between maximum and minimum brightness.
The general equation for a sine function is:
f(t) = A * sin(B * (t - C)) + D
In this case:
- A represents the amplitude, which is half of the total variation in brightness. As given, the brightness varies by ±0.30 magnitude, so the amplitude is 0.30 / 2 = 0.15.
- B represents the frequency and is given by B = 2π / T, where T is the time period. The time period between periods of maximum brightness is 5.6 days, so B = 2π / 5.6.
- C represents a phase shift, which determines the starting point of the sine function. We are told that at t = 0, the brightness is at its maximum, so C = 0.
- D represents the average brightness or magnitude of the star, which is given as 5.5.
Putting it all together, the function that models the brightness of the star is:
f(t) = 0.15 * sin((2π / 5.6)t) + 5.5
To model the brightness of the star as a function of time, we can use a sine function since it exhibits periodic behavior.
The general form of a sine function is given by:
y = A * sin(B * (x - C)) + D
Where:
- A represents the amplitude (the maximum variation in brightness)
- B represents the period (the time it takes to complete one full cycle)
- C represents the phase shift (the amount of time it takes before the star reaches its maximum brightness)
- D represents the vertical shift (the average or baseline brightness)
In this case, we can determine the values of A, B, C, and D using the given information:
- The amplitude is ±0.30 magnitude as the brightness varies by ±0.30 magnitude.
- The period is 5.6 days as it is the time between periods of maximum brightness.
- The phase shift is 0 since at t = 0 the brightness of the star is at its maximum.
Plugging these values into the sine function, we get:
y = ±0.30 * sin((2π / 5.6) * x) + 5.5
Therefore, the function that models the brightness of the star as a function of time, t, is:
brightness(t) = ±0.30 * sin((2π / 5.6) * t) + 5.5