A variable star is one whose brightness alternately increases and decreases. For one such star, the time between periods of maximum brightness is 4.7 days, the average brightness (or magnitude) of the star is 5.3, and its brightness varies by ±0.35 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 5.3 and that it is increasing.)
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To find a function that models the brightness of the star as a function of time, we can use a sine function since it follows a periodic pattern.
In this case, the time between periods of maximum brightness is 4.7 days. This corresponds to the period of the sine function.
The average brightness of the star is given as 5.3, which corresponds to the vertical shift of the sine function.
The brightness varies by ±0.35 magnitude, which corresponds to the amplitude of the sine function.
Putting it all together, the function that models the brightness of the star as a function of time, t, is:
f(t) = 5.3 + 0.35 * sin((2π/4.7) * t)
Where:
5.3 is the average brightness
0.35 is the amplitude
2π/4.7 is the angular frequency, which is determined by the period of 4.7 days
t is the time in days
To find a function that models the brightness of the star as a function of time, we need to consider the periodic nature of the star's brightness and the given information.
A common mathematical function used to model periodic phenomena is the sine or cosine function. Since the brightness alternately increases and decreases, we can use a cosine function to represent this behavior. Let's denote the function as f(t), where t represents time in days.
The general form of a cosine function is:
f(t) = A * cos(B(t - C)) + D
In this equation:
- A represents the amplitude, which is half the difference between the maximum and minimum brightness. In this case, it is ±0.35 divided by 2, so A = ±0.175.
- B represents the frequency of the oscillation. It can be calculated using the formula B = 2π / T, where T is the period between the maximum brightness. In this case, T = 4.7 days, so B ≈ 2π / 4.7.
- C represents the phase shift of the cosine wave. Since the brightness is increasing at t = 0, there is no phase shift, so C = 0.
- D represents the average brightness, which is given as 5.3.
Combining these values, the function that models the brightness of the star as a function of time can be written as:
f(t) = ±0.175 * cos((2π / 4.7) * t) + 5.3
Note: Remember that the amplitude (A) determines the range of the brightness variation, and the average brightness (D) determines the offset or average level of brightness.
This function will output the brightness of the star for any given time t.
period: 4.7, so 2π/k = 4.7 k = 2π/4.7
y = sin(2π/4.7 t)
average: 5.3, so
y = 5.3 + sin(2π/4.7 t)
brightness varies by ±0.35
y = 5.3 + 0.35 sin(2π/4.7 t)
(Assume that at t = 0 the brightness of the star is 5.3 and that it is increasing.)
No phase shift, so the above will work.