SOLUTION: The distance from the Sun can be approximated by the equation D=2.5cos[0.0172(n-185)+150, where D represents the distance in millions of kilometers and n represents the number of d

Question 948557: The distance from the Sun can be approximated by the equation D=2.5cos[0.0172(n-185)+150, where D represents the distance in millions of kilometers and n represents the number of days into a year.
a) How far will Earth be from the Sun on March 21st, the 80th day of the year?
This is what I did, but I don't think it's right.
D=2.5cos[0.0172(80-185)]+150
=2.5cos(-1.806)+150
=149.417....
then I have no clue how to find
b) The range of distance the Earth can be from the sun.
c) The period of the cycle, and what it means in context to the question.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
The distance from the Sun can be approximated by the equation D=2.5cos[0.0172(n-185)+150, where D represents the distance in millions of kilometers and n represents the number of days into a year.

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a) How far will Earth be from the Sun on March 21st, the 80th day of the year?

use the formula to find D.

Your calculator needs to be set to radians.

D = 2.5 * cos(0.0172(n-185)+150

D = distance in millions of kilometers.
n = the nth day of the year.

replace n with 80 in that formula and you get:

D = 2.5 * cos(0.0172 * (80-185)) + 150

you got D = 149.4173975 which I believe is a correct approximation based on some other references that I used to check if the answer was reasonable.

it is.

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b) The range of distance the Earth can be from the sun.

the range is from 147.5 to 152.5.

This can be found from the general equation for a cosine trig function.

that general equation is y = A * cos(B * (x-C)) + D

A is the amplitude.
B is the frequency.
C is the horizontal displacement.
D is the vertical displacement.

in your equation, the amplitude is 2.5 and the vertical displacement is 150.

the vertical displacement is the horizontal axis of the trig function.

the amplitude is plus or minus from the horizontal axis of the trig function.

since the axis is at y = 150, then the amplitude is at y = 150 plus or minus 2.5 which makes it y = from 147.5 to y = 152.5.

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c) The period of the cycle, and what it means in context to the question.

from the basic equation, B is the frequency.

in your equation B = .0172.

The frequency is equal to 2*pi / the period.

the period is equal to 2*pi divided by the frequency.

your period would therefore be equal to 2*pi / .0172 = 365.3014713

that's equivalent to the number of days in a year.

so your period is one year expressed in days.

the graph of your equation is shown below:

$$$

here's a reference that talks about the graph of the sine and cosine equations.

http://www.regentsprep.org/Regents/math/algtrig/ATT7/sinusoidal.htm

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