Question

Pluto's distance P(t)P(t)P, left parenthesis, t, right parenthesis (in billions of kilometers) from the sun as a function of time ttt (in years) can be modeled by a sinusoidal expression of the form a\cdot\sin(b\cdot t)+da⋅sin(b⋅t)+da, dot, sine, left parenthesis, b, dot, t, right parenthesis, plus, d. At year t=0t=0t, equals, 0, Pluto is at its average distance from the sun, which is 6.96.96, point, 9 billion kilometers. In 666666 years, it is at its closest point to the sun, which is 4.44.44, point, 4 billion kilometers away. Find P(t)P(t)P, left parenthesis, t, right parenthesis. \textit{t}tstart text, t, end text should be in radians.

Answer 1

Answer: P(t) = 1.25.sin($\frac{\pi}{3}$.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 = $\frac{2.\pi}{B}$

C is the phase shift or horizontal shift: $\frac{C}{B}$

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 = $\frac{largest - smallest}{2}$

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 = $\frac{6.9-4.4}{2}$

a = 1.25

b

A time period for Pluto is T=66 years:

66 = $\frac{2.\pi}{b}$

b = $\frac{\pi}{33}$

Vertical Shift

It can be calculated as:

d = $\frac{largest+smallest}{2}$

d = $\frac{6.9+4.4}{2}$

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($\frac{\pi}{3}$.t) + 5.65

The Pluto's distance from the sun as a function of time is

P(t) = 1.25.sin($\frac{\pi}{3}$.t) + 5.65