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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*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)=

Pluto's distance P(t) P(t) (in billions of kilometers) from the sun as a function of time t t (in years) can be modeled by a sinusoidal expression of the form a⋅sin⁡(b⋅t)+d a \cdot \sin (b \cdot t)+d .\newlineAt year t=0 t=0 , Pluto is at its average distance from the sun, which is 66.99 billion kilometers. In 6666 years, it is at its closest point to the sun, which is 44.44 billion kilometers away.\newlineFind P(t) P(t) .\newlinet t should be in radians.\newlineP(t)= P(t)=

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Q. Pluto's distance P(t) P(t) (in billions of kilometers) from the sun as a function of time t t (in years) can be modeled by a sinusoidal expression of the form a⋅sin⁡(b⋅t)+d a \cdot \sin (b \cdot t)+d .\newlineAt year t=0 t=0 , Pluto is at its average distance from the sun, which is 66.99 billion kilometers. In 6666 years, it is at its closest point to the sun, which is 44.44 billion kilometers away.\newlineFind P(t) P(t) .\newlinet t should be in radians.\newlineP(t)= P(t)=
  1. Given Distance Values: Since at t=0t=0, Pluto is at its average distance from the sun, dd equals the average distance, which is 6.96.9 billion kilometers.\newlined=6.9d = 6.9
  2. Calculating Amplitude: At t=66t=66 years, Pluto is at its closest point to the sun, which is 4.44.4 billion kilometers away. This means the amplitude aa is half the difference between the average distance and the closest distance.\newlinea=(6.9−4.4)/2a = (6.9 - 4.4) / 2\newlinea=2.5/2a = 2.5 / 2\newlinea=1.25a = 1.25
  3. Determining Minimum Point: Since Pluto is at its closest point to the sun at t=66t=66 years, this corresponds to the minimum point of the sinusoidal function, which occurs at −a-a. Therefore, the sinusoidal function must be negative at this point.
  4. Finding Period: To find bb, 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 6666 years, this is half the period. So the period TT is 66×2=13266 \times 2 = 132 years.
  5. Calculating b: The period T is related to b by the formula T=2πbT = \frac{2\pi}{b}. We solve for b using T=132T = 132.b=2πTb = \frac{2\pi}{T}b=2π132b = \frac{2\pi}{132}b=π66b = \frac{\pi}{66}
  6. Writing Sinusoidal Function: Now we have aa, bb, and dd. We can write the sinusoidal function P(t)P(t) as:\newlineP(t)=−1.25sin⁡(π66∗t)+6.9P(t) = -1.25\sin\left(\frac{\pi}{66} * t\right) + 6.9

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