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.

1. Find the formula of the trigonometric function that models Pluto's distance [tex]$D$[/tex] from the sun (in billion km) [tex]$t$[/tex] years after 2000. Define the function using radians.
[tex]\[
D(t)= \square
\][/tex]

2. How far will Pluto be from the sun in 2022? Round your answer, if necessary, to two decimal places.
[tex]\[
\square \text{ billion km}
\][/tex]



Answer :

Sure, let's go through the steps to find the trigonometric function and Pluto's distance from the sun in 2022.

### Step 1: Determine the period of Pluto's orbit
To find the period of the trigonometric function, we calculate the time between two successive perihelions. Given:
- Pluto's perihelion last occurred in 1989.
- Pluto's next perihelion will occur in 2237.

The period [tex]\( P \)[/tex] is:
[tex]\[ P = 2237 - 1989 = 248 \text{ years} \][/tex]

### Step 2: Calculate average distance and amplitude
Next, we find the average distance and the amplitude of the distance variation.

- Maximum distance (aphelion): 7.4 billion km
- Minimum distance (perihelion): 4.4 billion km

The average distance [tex]\( D_{\text{avg}} \)[/tex] is:
[tex]\[ D_{\text{avg}} = \frac{\text{Max distance} + \text{Min distance}}{2} = \frac{7.4 + 4.4}{2} = 5.9 \text{ billion km} \][/tex]

The amplitude [tex]\( A \)[/tex] is:
[tex]\[ A = \frac{\text{Max distance} - \text{Min distance}}{2} = \frac{7.4 - 4.4}{2} = 1.5 \text{ billion km} \][/tex]

### Step 3: Formulate the trigonometric function
The formula for Pluto's distance from the sun [tex]\( D(t) \)[/tex] can be modeled using the cosine function because the cosine function appropriately represents periodic variations. Given that [tex]\( t \)[/tex] is the time in years after 2000, we use the cosine function with period [tex]\( 248 \)[/tex] years:

[tex]\[ D(t) = \text{Average distance} + \text{Amplitude} \cdot \cos\left(\frac{2\pi t}{\text{Period}}\right) \][/tex]

Substitute the values:
[tex]\[ D(t) = 5.9 + 1.5 \cos\left(\frac{2\pi t}{248}\right) \][/tex]

### Step 4: Calculate Pluto's distance from the sun in 2022
To find the distance in 2022, we need [tex]\( t \)[/tex], which is the number of years from 2000 to 2022:

[tex]\[ t = 2022 - 2000 = 22 \][/tex]

Substitute [tex]\( t = 22 \)[/tex] into the trigonometric function:
[tex]\[ D(22) = 5.9 + 1.5 \cos\left(\frac{2\pi \cdot 22}{248}\right) \][/tex]

First, calculate the angle in radians:
[tex]\[ \frac{2\pi \cdot 22}{248} \approx 0.5574 \text{ radians} \][/tex]

Evaluate the cosine of this angle:
[tex]\[ \cos(0.5574) \approx 0.85 \][/tex]

Now, substitute this value back into the distance function:
[tex]\[ D(22) = 5.9 + 1.5 \cdot 0.85 = 5.9 + 1.275 = 7.175 \][/tex]

Rounding to two decimal places:
[tex]\[ D(22) = 7.17 \text{ billion km} \][/tex]

### Final Answers
The trigonometric function that models Pluto's distance [tex]\( D \)[/tex] from the sun [tex]\( t \)[/tex] years after 2000 is:
[tex]\[ D(t) = 5.9 + 1.5 \cos\left(\frac{2\pi t}{248}\right) \][/tex]

In the year 2022, Pluto will be approximately:
[tex]\[ 7.17 \text{ billion km} \][/tex]
from the sun.