While at sea level, suppose that you close the lid of an empty bottle... well, empty of any fluid besides air! The air inside is at the same pressure as the outside air, namely, $1$ standard atmosphere of pressure, or $1\mathrm{.}01105$ P a$.$ Suppose that the opening covered by the lid is a circle of radius $1\mathrm{.}25$ cm$.$

$($a$)$ What force does the air inside the bottle exert on the lid? What force does the air outside the bottle exert on the lid? What is the net force on the lid due to pressure, if any? Give the magnitude and direction for each force.

Suppose that you take this bottle with you in a car and you drive the parkway through Rocky Mountain National Park in Colorado, cresting a pass whose altitude is $10,000$ ft$.($Let$$s assume that the car is climate$-$controlled, so that the temperature doesn$$t change as you go from sea to summit.$)$ At that altitude, the outside air pressure is only $6\mathrm{.}97104$ P a$,$ but the pressure of the air inside the bottle is still the same atmospheric pressure at sea level. At $10,000$ ft:

$($b$)$ What force does the air inside the bottle exert on the lid? What force does the air outside the bottle exert on the lid? What is the net force on the lid due to pressure, if any? Give the magnitude and direction for each force.