Problem $2$: altitudes $(\backslash (\backslash$mathbf$\{20\}\backslash$mathbf$\{$~ p t s$\}\backslash )).$ Look up the radius of the earth and answer the following questions.

$($a$)(10$ pts$)$ Geopotential altitude is a measurement used in meteorology and aviation that represents the height of a point in the atmosphere relative to the Earth's mean sea level, adjusted for variations in gravity. Unlike geometric altitude, which is a straight$-$line measurement from sea level, geopotential altitude accounts for the fact that gravity changes with altitude. The relationship between geopotential altitude $(\backslash ($ h $\backslash ))$ and geometric altitude $\backslash (\backslash$left$($h$\_\{$G$\}\backslash$right$)\backslash )$ is given by

$\backslash [$

h$=\backslash$frac$\{$r$\}\{$r$+$h$\_\{$G$\}\}$ h$\_\{$G$\}$

$\backslash ]$

where $\backslash ($ r $\backslash )$ is the Earth radius. Engineers say that only at altitudes above $65$ km does the difference between the geometric and geopotential altitudes exceed $\backslash (1\backslash \%\backslash ).$ Calculate the exact value of the geometric altitude at which this difference is precisely $\backslash (1\backslash \%\backslash ).$

$($b$)(10$ pts$)$ For the flight of airplanes in the earth's atmosphere, the variation of the acceleration of gravity with altitude is generally ignored. One of the highest$-$flying aircraft has been the Lockheed U$-2$ which was designed to cruise at $\backslash (70,000\backslash$mathrm$\{$ft$\}\backslash ).$ How much does the acceleration of gravity at this altitude differ from the value at sea level?