The atmosphere of Mars exerts a pressure of only 600\. Pa on the surface and has a density of only \(0.0200 \mathrm{~kg} / \mathrm{m}^{3}\). a) What is the thickness of the Martian atmosphere, assuming the boundary between atmosphere and outer space to be the point where atmospheric pressure drops to \(0.0100 \%\) of its yalue at surface level? b) What is the atmospheric pressure at the bottom of Mars's Hellas Planitia canyon, at a depth of \(7.00 \mathrm{~km} ?\) c) What is the atmospheric pressure at the top of Mars's Olympus Mons volcano, at a height of \(27.0 \mathrm{~km} ?\) d) Compare the relative change in air pressure, \(\Delta p / p\), between these two points on Mars and between the equivalent extremes on Earth-the Dead Sea shore, at \(400 . \mathrm{m}\) below sea level, and Mount Everest, at an altitude of \(8850 \mathrm{~m}\).

#a) Finding the thickness of Martian atmosphere# #tag_title#Step 3: Determine the constant α#tag_content#Using the surface pressure and surface density provided, we can solve for the constant α: α = \(\frac{p_0}{\rho_0 c}\). #tag_title#Step 4: Calculate the thickness of the Martian atmosphere#tag_content#Now that we have the constant α, we can plug it into the equation from Step 2 to determine the height \(h_A\): \(h_A = \frac{1}{\alpha} \ln \frac{p_0}{0.0001p_0}\). b) Calculating the atmospheric pressure at the bottom of Hellas Planitia #tag_title#Step 1: Calculate the height difference#tag_content#The depth of Hellas Planitia, \(h_B\), is approximately 7 km. Hence, the height difference is -7 km (since it is below the surface). #tag_title#Step 2: Calculate the atmospheric pressure at the bottom of Hellas Planitia#tag_content#Using the relationship we derived in a), we can determine the pressure at the bottom of Hellas Planitia: \(p(h_B) = p_0 \cdot \mathrm{exp}(- \alpha h_B)\). c) Calculating the atmospheric pressure at the top of Olympus Mons #tag_title#Step 1: Calculate the height difference#tag_content#The height of Olympus Mons, \(h_C\), is approximately 21 km. #tag_title#Step 2: Calculate the atmospheric pressure at the top of Olympus Mons#tag_content#Using the relationship we derived in a), we can determine the pressure at the top of Olympus Mons: \(p(h_C) = p_0 \cdot \mathrm{exp}(- \alpha h_C)\). d) Comparing relative change in air pressure on Mars and Earth #tag_title#Step 1: Calculate the relative change in air pressure for Mars#tag_content#From part a), we calculated the pressure at the bottom of Hellas Planitia and the top of Olympus Mons. To determine the relative change in air pressure on Mars, we can use the formula: \(\frac{p(h_C) - p(h_B)}{p_0}\). #tag_title#Step 2: Calculate the relative change in air pressure for Earth#tag_content#Using the known values for Earth's atmosphere, calculate the relative change in air pressure from the highest point (Mount Everest) to the lowest point (Mariana Trench). \(\frac{p_{Everest} - p_{Mariana}}{p_{0, Earth}}\). #tag_title#Step 3: Compare the relative changes in air pressure#tag_content#Calculate the ratio of the relative change in air pressure between Mars and Earth: \(\frac{(\frac{p(h_C) - p(h_B)}{p_0})}{(\frac{p_{Everest} - p_{Mariana}}{p_{0, Earth}})}\).