College

The gravitational force \( F \) between an object and the Earth is inversely proportional to the square of the distance from the object to the center of the Earth.

If an astronaut weighs 228 pounds on the surface of the Earth, what will the astronaut weigh 200 miles above the Earth? Assume that the radius of the Earth is 4000 miles.

Answer :

Check the picture below.

[tex]\qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\ \textit{\underline{x} varies inversely with }\underline{z^5} ~\hspace{5.5em} \stackrel{\textit{constant of variation}}{x=\cfrac{\stackrel{\downarrow }{k}}{z^5}~\hfill } \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{"F" inversely proportional with }"d^2"}{ {\large \begin{array}{llll} F=\cfrac{k}{d^2} \end{array}}} ~~ \textit{we also know} \begin{cases} F=228\\ d=4000 \end{cases} \implies 228=\cfrac{k}{4000^2} \\\\\\ 228(4000^2)=k\implies 3648000000=k~\hfill \boxed{F=\cfrac{3648000000}{d^2}} \\\\\\ \textit{when d = 200, what is "F"?}\qquad F=\cfrac{3648000000}{4200^2}\implies F\approx 206.8~lbs[/tex]

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