Chapter 2.2, Problem 22E

Viewing Earth: Astronauts looking at Earth from a spacecraft can see only a portion of the surface. See figure 2.47. The fraction F of the surface of Earth that is visible at a height h, in kilometers, above the surface is given by the formula

$F=\frac{0.5h}{R+h}$

Here R is the radius of Earth, about 6380 kilometers. (for comparison, 1 kilometer is about 0.62 mile, and the moon is about 380,000 kilometers from Earth.)

a. Make a graph of F versus h covering heights up to 100,000 kilometers.

b. A value of F equal to 0.25 means that 25%, or one-quarter, of Earth’s surface is visible. At what height is this fraction visible?

c. During one flight of a space shuttle, astronauts performed an extravehicular activity at a height of 280 kilometers. What fraction of the surface of Earth is visible at that height?

d. Is the graph of F concave up or concave down? Explain your answer in practical terms.

e. Determine the limiting value for F as the height h gets larger. Explain your answer in practical terms.