Answer: P(t) = 1.25.sin([tex]\frac{\pi}{3}[/tex].t) + 5.65
Step-by-step explanation: A motion repeating itself in a fixed time period is a periodic motion and can be modeled by the functions:
y = A.sin(B.t - C) + D or y = Acos(B.t - C) + D
where:
A is amplitude A=|A|
B is related to the period by: T = [tex]\frac{2.\pi}{B}[/tex]
C is the phase shift or horizontal shift: [tex]\frac{C}{B}[/tex]
D is the vertical shift
In this question, the motion of Pluto is modeled by a sine function and doesn't have phase shift, C = 0.
Amplitude:
a = [tex]\frac{largest - smallest}{2}[/tex]
At t=0, Pluto is the farthest from the sun, a distance 6.9 billions km away. At t=66, it is closest to the star, P(66) = 4.4 billions km. Then:
a = [tex]\frac{6.9-4.4}{2}[/tex]
a = 1.25
b
A time period for Pluto is T=66 years:
66 = [tex]\frac{2.\pi}{b}[/tex]
b = [tex]\frac{\pi}{33}[/tex]
Vertical Shift
It can be calculated as:
d = [tex]\frac{largest+smallest}{2}[/tex]
d = [tex]\frac{6.9+4.4}{2}[/tex]
d = 5.65
Knowing a, b and d, substitute in the equivalent positions and find P(t).
P(t) = a.sin(b.t) + d
P(t) = 1.25.sin([tex]\frac{\pi}{3}[/tex].t) + 5.65
The Pluto's distance from the sun as a function of time is
P(t) = 1.25.sin([tex]\frac{\pi}{3}[/tex].t) + 5.65
