Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function. $\newline$ 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 $2237$ . $\newline$ Find the formula of the trigonometric function that models Pluto's distance $D$ from the sun (in billion $\mathrm{km}$ ) $t$ years after $2000$ . Define the function using radians. $\newline$ $D(t)=\square$ $\newline$ How far will Pluto be from the sun in $2022$ ? Round your answer, if necessary, to two decimal places. $\newline$ billion $\mathrm{km}$

View step-by-step help

Home

Math Problems

Algebra 1

Interpret parts of quadratic expressions: word problems

Full solution

Q. Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function.

\newline

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

\newline

Find the formula of the trigonometric function that models Pluto's distance

D

from the sun (in billion

\mathrm{km}

)

t

years after

2000

. Define the function using radians.

\newline

D(t)=\square

\newline

How far will Pluto be from the sun in

2022

? Round your answer, if necessary, to two decimal places.

\newline

billion

\mathrm{km}

Determine Amplitude: Determine the amplitude of the trigonometric function. $\newline$ The amplitude is half the distance between the maximum and minimum values. $\newline$ Amplitude = $(\text{Maximum distance} - \text{Minimum distance}) / 2$ $\newline$ Amplitude = $(7.4 \text{ billion km} - 4.4 \text{ billion km}) / 2$ $\newline$ Amplitude = $3 \text{ billion km}$
Determine Vertical Shift: Determine the vertical shift of the trigonometric function. $\newline$ The vertical shift is the average of the maximum and minimum values. $\newline$ Vertical shift = $(\text{Maximum distance} + \text{Minimum distance}) / 2$ $\newline$ Vertical shift = $(7.4 \text{ billion km} + 4.4 \text{ billion km}) / 2$ $\newline$ Vertical shift = $5.9 \text{ billion km}$
Determine Period: Determine the period of the trigonometric function. $\newline$ The period is the time it takes for Pluto to go from one perihelion to the next. $\newline$ Period $=$ Next perihelion year $-$ Last perihelion year $\newline$ Period $=$ $2237$ $-$ $1989$ $\newline$ Period $=$ $248$ years
Convert to Radians: Convert the period into radians since we are defining the function using radians. $\newline$ The period in radians for a cosine function is $2\pi$ , so we need to find the value that corresponds to $248$ years. $\newline$ Period in radians = $\frac{2\pi}{248}$
Determine Horizontal Shift: Determine the horizontal shift of the trigonometric function. $\newline$ The horizontal shift corresponds to the year Pluto last reached perihelion relative to the year $2000$ . $\newline$ Horizontal shift $= \text{Last perihelion year} - 2000$ $\newline$ Horizontal shift $= 1989 - 2000$ $\newline$ Horizontal shift $= -11$ years
Write Trigonometric Formula: Write the formula for the trigonometric function. $\newline$ We will use a cosine function because it starts at a maximum, and we know Pluto was at perihelion in $1989$ , which is a minimum point. $\newline$ $D(t) = \text{Amplitude} \times \cos(\text{Period in radians} \times (t - \text{Horizontal shift})) + \text{Vertical shift}$ $\newline$ $D(t) = 3 \times \cos(2\pi / 248 \times (t + 11)) + 5.9$
Calculate Distance in $2022$ : Calculate Pluto's distance from the sun in $2022$ . $\newline$ $t = 2022 - 2000$ $\newline$ $t = 22$ years $\newline$ $D(22) = 3 \cdot \cos\left(\frac{2\pi}{248} \cdot (22 + 11)\right) + 5.9$ $\newline$ $D(22) = 3 \cdot \cos\left(\frac{2\pi}{248} \cdot 33\right) + 5.9$
Perform Cosine Calculation: Perform the calculation for the cosine term. $\newline$ $D(22) = 3 \times \cos(\frac{2\pi}{248} \times 33) + 5.9$ $\newline$ $D(22) = 3 \times \cos(\frac{2\pi \times 33}{248}) + 5.9$ $\newline$ $D(22) = 3 \times \cos(0.4188790204786391) + 5.9$ $\newline$ $D(22) \approx 3 \times \cos(0.4188790204786391) + 5.9$
Use Calculator for Cosine: Use a calculator to find the cosine value and complete the calculation. $\newline$ $D(22) \approx 3 \times 0.9170605047794995 + 5.9$ $\newline$ $D(22) \approx 2.7511815143384985 + 5.9$ $\newline$ $D(22) \approx 8.651181514338498$ $\newline$ Round to two decimal places. $\newline$ $D(22) \approx 8.65$ billion km