To 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, and to determine how far Pluto will be from the Sun in 2022, we'll proceed as follows:
### Step 1: Finding the Trigonometric Function
1. Identify the parameters given:
- Minimum distance (perihelion): [tex]\( D_{\min} = 4.4 \)[/tex] billion kilometers.
- Pluto reached perihelion in 1989.
- Pluto will next reach perihelion in 2237.
2. Calculate the period of Pluto’s orbit:
- The period [tex]\( T \)[/tex] is the time between two consecutive perihelion passages.
- [tex]\( T = 2237 - 1989 = 248 \)[/tex] years.
3. Determine the angular frequency:
- Angular frequency [tex]\( \omega = \frac{2\pi}{T} = \frac{2\pi}{248} \)[/tex].
4. Formulate the distance function:
- Given that perihelion (minimum distance) occurs at a cosine maximum (i.e., [tex]\(\cos(0) = 1\)[/tex]), we can model the distance [tex]\(D(t)\)[/tex] using the cosine function.
- Let [tex]\( t \)[/tex] be the number of years after 2000.
- The function would be:
[tex]\[
D(t) = D_{\min} \cdot \cos\left(\omega (t - (2000 - 1989))\right)
\][/tex]
- Since [tex]\( 2000 - 1989 = 11 \)[/tex], we substitute and get:
[tex]\[
D(t) = 4.4 \cdot \cos\left(\frac{2\pi}{248}(t - 11)\right)
\][/tex]
So, the formula for Pluto's distance from the Sun [tex]\( D(t) \)[/tex] years after 2000 is:
[tex]\[
D(t) = 4.4 \cdot \cos\left(\frac{2\pi}{248}(t - 11)\right)
\][/tex]
### Step 2: Calculate Pluto’s Distance in 2022
1. Substitute [tex]\( t = 2022 - 2000 = 22 \)[/tex] into the formula:
[tex]\[
D(22) = 4.4 \cdot \cos\left(\frac{2\pi}{248}(22 - 11)\right)
\][/tex]
2. Simplify the argument of the cosine function:
[tex]\[
D(22) = 4.4 \cdot \cos\left(\frac{2\pi}{248} \times 11\right)
\][/tex]
3. Calculate the value:
- Using the trigonometric function,
[tex]\[
D(22) \approx 2.95 \text{ billion km}
\][/tex]
Therefore, Pluto will be approximately [tex]\( 2.95 \)[/tex] billion kilometers away from the Sun in 2022.
### Final Answer
The trigonometric function that models Pluto's distance from the Sun [tex]\( D(t) \)[/tex] years after 2000 is:
[tex]\[
D(t) = 4.4 \cdot \cos\left(\frac{2\pi}{248}(t - 11)\right)
\][/tex]
Pluto will be approximately [tex]\( 2.95 \)[/tex] billion kilometers away from the Sun in 2022.
