Question 181207
>>...weight...varies inversely as the 
square of the distance from the earth's center...<<
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That says:

{{{matrix(1,3,W, "=", k/d^2)}}}

>>...a 450 kg object weighs 4410 Newtons 
at the surface of the earth...<<

On the surface of the earth the distance d 
to the center of the earth is 6500 km, so
when d = 6500, then W = 4410. We substitute
that in the equation:

{{{matrix(1,3,W, "=", k/d^2)}}}
{{{matrix(1,3,4410, "=", k/6500^2)}}}
{{{matrix(1,3,4410, "=", k/42250000)}}}

Now we can solve for k. So we multiply both 
sides by {{{42250000}}}.

{{{matrix(1,3,4410*42250000,"=",k)}}}

{{{matrix(1,3,186322500000,"=", k)}}}

So we substitute {{{186322500000}}} for {{{k}}} in

{{{matrix(1,3,W, "=", k/d^2)}}}

{{{matrix(1,3,W, "=", 186322500000/d^2)}}}

Now we read the question:
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>>...What does a 450 kg object weigh 500 km 
away for the earths surface?...<<
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Since it's 500 km above the earth and it's 6500 km 
from the surface down to center of the earth, 
then the object is 500+6500 or 7000 km from 
the center of the earth.  So when d = 7000km,
we want to know what W is when d is 7000.

{{{matrix(1,3,W, "=", 186322500000/d^2)}}}
{{{matrix(1,3,4410, "=", 186322500000/7000^2)}}} 
{{{matrix(1,3,W, "=", 186322500000/7000^2)}}}
{{{matrix(1,3,W, "=", 186322500000/49000000)}}}
{{{matrix(1,3,W, "=", 3802.5)}}}

So it weighs 3802.5 Newtons.

Edwin</pre>