Two satellites are orbiting earth. The path of one has the equation x^2 +y^2 = 2250000. The orbit of the other is 200 km farther from the center of earth. In one orbit, how much farther does the second satellite travel than the first one?
Two satellites are orbiting earth. The path of one has the equation x^2 +y^2 = 2250000. The orbit of the other is 200 km farther from the center of earth. In one orbit, how much farther does the second satellite travel than the first one?
The equation of the satellite's orbit is given in the form:
x^{2} + y^{2} = r^{2}
Which is the equation of a circle, where r is the radius. Since the equation is:
x^{2} +y^{2} = 2250000
r^{2} is therefore 2250000, so take the square root of it to find r
r^{2} = 2250000
r = √2250000 = 1500km
As Robert showed, you don't actually need to find the radius in this case, but you can use it if you like. Since the second satellite's radius is given by (r+200)km, to find how much further the second satellite travels, subtract the circumference of the first satellite from the circumference of the second:
2π(r+200) - 2πr = 2π(r+200-r) [I took out a common factor of 2π from both terms]
= 2π*200 = 400π km
If you did this using the radii, you'd do it like so:
2π*1700 - 2π*1500 = 2π(1700-1500) [again, I took out a common factor of 2π]
= 2π*200 = 400π km
So, you can see we get the same answer regardless of whether we actually use 1700km and 1500km, or (r+200) and r as the radii.
If the first radius is r km, the other radius is (r+200) km.
In one orbit, the second one travels 2pi(r+200-r) = 400 pi km farther than the first one.
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Attn: To answer the question, you don't need the information x^{2} +y^{2} = 2250000.
x^{2} + y^{2} = r^{2} is the equation for a circle. So for your problem, take the square root of 2250000 to get the radius, in this case, 1500. I am assuming km is the units.
if the orbit of the other is 200km farther out, the radius is 200km greater, so it's radius is 1700km.
you then figure out the distance each travels in the orbit (circumference) c=2rpi,
first satellite c=2(1500)(3.14)=9420km
second satellite c=2(1700)(3.14)=10676km
just subtract the distance the first one travels from the distance the second one travels to get your answer.
If you didn't have the radius of the first satellite's orbit, you could set up an equation as follows: 2(r+200)pi -2rpi =x circumference of larger - circumference of smaller = distance larger travels farther
which reduces to 2rpi + 400pi -2rpi = x
which reduces to 400pi = x
your answer would be the same either way.
hope this helps!
Comments
I know r is the radius, but what is the radius?
r = sqrt(2250000) = 1500 km
r^{2} = 2250000
r = 1500 km