plutos distance p(t) (in billions of kilometers) from the sun as a function of time t (in years) can be…

plutos distance p(t) (in billions of kilometers) from the sun as a function of time t (in years) can be modeled by a sinusoidal expression of the form a·sin(b·t)+d. at year t = 0, pluto is at its average distance from the sun, which is 6.9 billion kilometers. in 66 years, it is at its closest point to the sun, which is 4.4 billion kilometers away. find p(t), t should be in radians. p(t) =

plutos distance p(t) (in billions of kilometers) from the sun as a function of time t (in years) can be modeled by a sinusoidal expression of the form a·sin(b·t)+d. at year t = 0, pluto is at its average distance from the sun, which is 6.9 billion kilometers. in 66 years, it is at its closest point to the sun, which is 4.4 billion kilometers away. find p(t), t should be in radians. p(t) =

Answer

Explanation:

Step1: Determine the value of $d$

The average value of the sinusoidal - function is $d$. Given that at $t = 0$, $P(0)=d = 6.9$ billion kilometers.

Step2: Determine the value of $a$

The amplitude $a$ is half of the difference between the maximum and minimum values. The maximum value is the average value plus the amplitude, and the minimum value is the average value minus the amplitude. Given the minimum value is $4.4$ billion kilometers and the average value $d = 6.9$ billion kilometers. So, $a=6.9 - 4.4=2.5$ billion kilometers.

Step3: Determine the value of $b$

The period $T$ of a sinusoidal function $y = a\sin(bt)+d$ is related to $b$ by the formula $T=\frac{2\pi}{b}$. We know that the time from the average - position to the closest - position is half of the period. If it takes 66 years to go from the average position to the closest position, then the period $T = 4\times66=264$ years. Using the formula $T=\frac{2\pi}{b}$, we can solve for $b$: $b=\frac{2\pi}{T}=\frac{2\pi}{264}=\frac{\pi}{132}$.

Answer:

$P(t)=2.5\sin\left(\frac{\pi}{132}t\right)+6.9$