Suppose you are k miles (not feet) above the ground. The radius of the earth is approximat...

Suppose you are k miles (not feet) above the ground. The radius of the earth is approximately 3960 miles. At the point where your line of sight meets the earth, it is perpendicular to the radius of the earth, as shown in the figure.

(a) Use the Pythagorean Theorem (see the Geometry Review Appendix) to show that

 

---

(b) Show that the equation in part (a) simplifies to

 

---

(c) If you are h feet above the ground, then you are h/5280 miles high (why?). Use this fact and the equation in part (b) to obtain the formula used in Example 1.

EXAMPLE 1

Suppose you are located h feet above the ground. Because of the curvature of the earth, the maximum distance you can see is approximately d miles, where

 .

How far can you see from the 500-foot-high Smith Tower in Seattle and from the 1454-foot-high Sears Tower in Chicago?

SOLUTION

For the Smith Tower, substitute 500 for h in the formula, and use your calculator:

 .

For the Sears Tower, you can see almost 47 miles because

 .