SOLUTION: The time a planet takes to revolve once around the sun relates directly to the {{{3/2}}} power of the planet's average distance from the sun in millions of kilometres. Thus: {{{t=f

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Question 876382: The time a planet takes to revolve once around the sun relates directly to the 3%2F2 power of the planet's average distance from the sun in millions of kilometres. Thus: t=f%28d%29=kd%5E%283%2F2%29
a) if it takes the earth on year to revolve around the sun at an average distance of 150 000 000 km, determine the value of k. Write the equation for f(d).
b) Use f(d) to find how many earth years it takes Mercury to revolve once around the sun, if its average distance from the sun is 58 000 000km.
c) Is this an irrational function? Does it have a restricted domain?
thanks
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
t= time a planet takes to revolve once around the sun (in Earth years).
d= distance from the sun to the planet (in millions of km).
t=f%28d%29=kd%5E%28%223+%2F+2%22%29

a) For Earth, t=1 and d=150
1=k%2A150%5E%28%223+%2F+2%22%29
1%2F150%5E%28%223+%2F+2%22%29=k
k=1%2F750sqrt%286%29--->highlight%28k=sqrt%286%29%2F4500%29
The approximate value is highlight%28k=0.00054433%29
highlight%28t=f%28d%29=0.00054433d%5E%28%223+%2F+2%22%29%29

b) For Mercury, d=58 , so
t=f%2858%29=0.00054433%2A58%5E%28%223+%2F+2%22%29=0.240
It takes Mercury highlight%280.240%29 Earth years to revolve once around the sun.

c) I had not heard the term "irrational function" before, but then again, I am not a fan of calling anything names.
t=f%28d%29=0.00054433d%5E%28%223+%2F+2%22%29 is not a rational function,
because rational functions are quotients of polynomials, and t=f%28d%29=0.00054433d%5E%28%223+%2F+2%22%29 is an exponential function.
It does have a restricted domain because we expect d%3E0 ,
and d%3C0 makes d%5E%28%223+%2F+2%22%29=sqrt%28d%5E3%29 undefined.