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Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function. Pluto's maximum distance from the sun (aphelion) is 7.4 billion kilometers. Its minimum distance from the sun (perihelion) is 4.4 billion kilometers. Pluto last reached its perihelion in the year 1989 , and will next reach its perihelion in 2237. Find the formula of the trigonometric function that models Pluto's distance D from the sun (in billion km ) t years after 2000 . Define the function using radians. How far will Pluto be from the sun in 2022? Round your answer, if necessary, to two decimal places. billion km

Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function.\newlinePluto's maximum distance from the sun (aphelion) is 77.44 billion kilometers. Its minimum distance from the sun (perihelion) is 44.44 billion kilometers. Pluto last reached its perihelion in the year 19891989 , and will next reach its perihelion in 22372237.\newlineFind the formula of the trigonometric function that models Pluto's distance D D from the sun (in billion km \mathrm{km} ) t t years after 20002000 . Define the function using radians.\newlineD(t)=□ D(t)=\square \newlineHow far will Pluto be from the sun in 20222022 ? Round your answer, if necessary, to two decimal places.\newlinebillion km \mathrm{km}

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Q. Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function.\newlinePluto's maximum distance from the sun (aphelion) is 77.44 billion kilometers. Its minimum distance from the sun (perihelion) is 44.44 billion kilometers. Pluto last reached its perihelion in the year 19891989 , and will next reach its perihelion in 22372237.\newlineFind the formula of the trigonometric function that models Pluto's distance D D from the sun (in billion km \mathrm{km} ) t t years after 20002000 . Define the function using radians.\newlineD(t)=□ D(t)=\square \newlineHow far will Pluto be from the sun in 20222022 ? Round your answer, if necessary, to two decimal places.\newlinebillion km \mathrm{km}
  1. Determine Amplitude: Determine the amplitude of the trigonometric function.\newlineThe amplitude is half the distance between the maximum and minimum values.\newlineAmplitude = (Maximum distance−Minimum distance)/2(\text{Maximum distance} - \text{Minimum distance}) / 2\newlineAmplitude = (7.4 billion km−4.4 billion km)/2(7.4 \text{ billion km} - 4.4 \text{ billion km}) / 2\newlineAmplitude = 3 billion km3 \text{ billion km}
  2. Determine Vertical Shift: Determine the vertical shift of the trigonometric function.\newlineThe vertical shift is the average of the maximum and minimum values.\newlineVertical shift = (Maximum distance+Minimum distance)/2(\text{Maximum distance} + \text{Minimum distance}) / 2\newlineVertical shift = (7.4 billion km+4.4 billion km)/2(7.4 \text{ billion km} + 4.4 \text{ billion km}) / 2\newlineVertical shift = 5.9 billion km5.9 \text{ billion km}
  3. Determine Period: Determine the period of the trigonometric function.\newlineThe period is the time it takes for Pluto to go from one perihelion to the next.\newlinePeriod == Next perihelion year −- Last perihelion year\newlinePeriod == 22372237 −- 19891989\newlinePeriod == 248248 years
  4. Convert to Radians: Convert the period into radians since we are defining the function using radians.\newlineThe period in radians for a cosine function is 2π2\pi, so we need to find the value that corresponds to 248248 years.\newlinePeriod in radians = 2π248\frac{2\pi}{248}
  5. Determine Horizontal Shift: Determine the horizontal shift of the trigonometric function.\newlineThe horizontal shift corresponds to the year Pluto last reached perihelion relative to the year 20002000.\newlineHorizontal shift =Last perihelion year−2000= \text{Last perihelion year} - 2000\newlineHorizontal shift =1989−2000= 1989 - 2000\newlineHorizontal shift =−11= -11 years
  6. Write Trigonometric Formula: Write the formula for the trigonometric function.\newlineWe will use a cosine function because it starts at a maximum, and we know Pluto was at perihelion in 19891989, which is a minimum point.\newlineD(t)=Amplitude×cos⁡(Period in radians×(t−Horizontal shift))+Vertical shiftD(t) = \text{Amplitude} \times \cos(\text{Period in radians} \times (t - \text{Horizontal shift})) + \text{Vertical shift}\newlineD(t)=3×cos⁡(2π/248×(t+11))+5.9D(t) = 3 \times \cos(2\pi / 248 \times (t + 11)) + 5.9
  7. Calculate Distance in 20222022: Calculate Pluto's distance from the sun in 20222022.\newlinet=2022−2000t = 2022 - 2000\newlinet=22t = 22 years\newlineD(22)=3⋅cos⁡(2π248⋅(22+11))+5.9D(22) = 3 \cdot \cos\left(\frac{2\pi}{248} \cdot (22 + 11)\right) + 5.9\newlineD(22)=3⋅cos⁡(2π248⋅33)+5.9D(22) = 3 \cdot \cos\left(\frac{2\pi}{248} \cdot 33\right) + 5.9
  8. Perform Cosine Calculation: Perform the calculation for the cosine term.\newlineD(22)=3×cos⁡(2π248×33)+5.9D(22) = 3 \times \cos(\frac{2\pi}{248} \times 33) + 5.9\newlineD(22)=3×cos⁡(2π×33248)+5.9D(22) = 3 \times \cos(\frac{2\pi \times 33}{248}) + 5.9\newlineD(22)=3×cos⁡(0.4188790204786391)+5.9D(22) = 3 \times \cos(0.4188790204786391) + 5.9\newlineD(22)≈3×cos⁡(0.4188790204786391)+5.9D(22) \approx 3 \times \cos(0.4188790204786391) + 5.9
  9. Use Calculator for Cosine: Use a calculator to find the cosine value and complete the calculation.\newlineD(22)≈3×0.9170605047794995+5.9D(22) \approx 3 \times 0.9170605047794995 + 5.9\newlineD(22)≈2.7511815143384985+5.9D(22) \approx 2.7511815143384985 + 5.9\newlineD(22)≈8.651181514338498D(22) \approx 8.651181514338498\newlineRound to two decimal places.\newlineD(22)≈8.65D(22) \approx 8.65 billion km

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