Pluto's 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 $a \cdot \sin (b \cdot t)+d$ . $\newline$ 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. $\newline$ Find $P(t)$ . $\newline$ $t$ should be in radians. $\newline$ $P(t)=$

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Q. Pluto's 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 \cdot \sin (b \cdot t)+d

\newline

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.

\newline

Find

P(t)

\newline

t

should be in radians.

\newline

P(t)=

Given Distance Values: Since at $t=0$ , Pluto is at its average distance from the sun, $d$ equals the average distance, which is $6.9$ billion kilometers. $\newline$ $d = 6.9$
Calculating Amplitude: At $t=66$ years, Pluto is at its closest point to the sun, which is $4.4$ billion kilometers away. This means the amplitude $a$ is half the difference between the average distance and the closest distance. $\newline$ $a = (6.9 - 4.4) / 2$ $\newline$ $a = 2.5 / 2$ $\newline$ $a = 1.25$
Determining Minimum Point: Since Pluto is at its closest point to the sun at $t=66$ years, this corresponds to the minimum point of the sinusoidal function, which occurs at $-a$ . Therefore, the sinusoidal function must be negative at this point.
Finding Period: To find $b$ , we need to know that the period of the sinusoidal function is the time it takes for Pluto to complete one full orbit. Since it's at its closest point at $66$ years, this is half the period. So the period $T$ is $66 \times 2 = 132$ years.
Calculating b: The period T is related to b by the formula $T = \frac{2\pi}{b}$ . We solve for b using $T = 132$ . $b = \frac{2\pi}{T}$ $b = \frac{2\pi}{132}$ $b = \frac{\pi}{66}$
Writing Sinusoidal Function: Now we have $a$ , $b$ , and $d$ . We can write the sinusoidal function $P(t)$ as: $\newline$ $P(t) = -1.25\sin\left(\frac{\pi}{66} * t\right) + 6.9$