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. D(t)=◻ How far will Pluto be from the sun in 2022 ? Round your answer, if necessary, to two decimal places. ◻ billion km

Question

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. D(t)=◻ How far will Pluto be from the sun in 2022 ? Round your answer, if necessary, to two decimal places. ◻ billion km

Bytelearn · Accepted Answer

Determine Amplitude: First, we need to determine the amplitude of the trigonometric function. The amplitude is half the distance between the maximum and minimum values. Amplitude = (Maximum distance - Minimum distance) / 2 Amplitude = (7.4 billion km - 4.4 billion km) / 2 Amplitude = 3 billion km / 2 Amplitude = 1.5 billion kmCalculate Vertical Shift: Next, we calculate the vertical shift, which is the average of the maximum and minimum distances. Vertical shift = (Maximum distance + Minimum distance) / 2 Vertical shift = (7.4 billion km + 4.4 billion km) / 2 Vertical shift = 11.8 billion km / 2 Vertical shift = 5.9 billion kmFind Period: Now, we need to find the period of the function. The period is the time it takes for Pluto to go from one perihelion to the next. Period = Next perihelion year - Last perihelion year Period = 2237 - 1989 Period = 248 yearsConvert to Radians: To define the function using radians, we need to convert the period into radians. Since the period is the time for a full cycle, it corresponds to 2π radians. Radians per year = 2π / Period Radians per year = 2π / 248Write Trigonometric Function: We can now write the trigonometric function for Pluto's distance from the sun. We will use a cosine function, which starts at a maximum value at t=0 (year 2000). We need to adjust the phase shift to account for the fact that the last perihelion was in 1989, which is 11 years before 2000. Phase shift = Radians per year * (2000 - Last perihelion year) Phase shift = (2π / 248) * (2000 - 1989) Phase shift = (2π / 248) * 11Calculate D(22): The trigonometric function that models Pluto's distance from the sun t years after 2000 is: D(t) = Amplitude * cos(Radians per year * t - Phase shift) + Vertical shift D(t) = 1.5 billion km * cos((2π / 248) * t - (2π / 248) * 11) + 5.9 billion kmPerform Calculation: To find how far Pluto will be from the sun in 2022, we need to calculate D(t) for t = 2022 - 2000. t = 2022 - 2000 t = 22 years D(22) = 1.5 billion km * cos((2π / 248) * 22 - (2π / 248) * 11) + 5.9 billion kmFinal Result: Now we perform the calculation for D(22): D(22) = 1.5 billion km * cos((2π / 248) * 22 - (2π / 248) * 11) + 5.9 billion km D(22) ≈ 1.5 billion km * cos(0.55850536 - 0.27925268) + 5.9 billion km D(22) ≈ 1.5 billion km * cos(0.27925268) + 5.9 billion km D(22) ≈ 1.5 billion km * 0.9605305 + 5.9 billion km D(22) ≈ 1.44079575 billion km + 5.9 billion km D(22) ≈ 7.34079575 billion km Rounded to two decimal places, Pluto will be approximately 7.34 billion km from the sun in 2022.

Full solution

More problems from Find a value using two-variable equations: word problems

Related topics