Question 4790: How can I solve the following problem: The tallest inhabited building in the world is the Sears Tower in Chicago. If the observation tower is 1450 feet above ground level, how far can a person standing in the observation tower see (with the aid of a telescope)? Use 3960 miles for the radius of Earth.
Note: 1 mile = 5280 feet
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! You can solve this problem as follows:
Draw a circle that represents the earth.
Draw a radius and extend this beyond the circumference of the circle. This extension will represent Sears Tower in Chicago.
From the top of the "tower" in your diagram. draw a line segment so that it is tangent to the circle.
Draw another radius from the circle centre to the point of tangency.
These two line segments (the radius and the tangent line segment from the top of the "tower") are perpendicular.
You now have a right triangle in your diagram whose base is the second radius and whose height is the tangent line segment from the top of the "tower" to the point of tangency. The hypotenuse of this right triangle is the first radius plus the extension representing the Sears Tower in Chicago.
Having done all of this graphic arts you are ready to solve the problem.
You can probably see that the distance a person can "see" from the top of the Sears Tower in Chicago is represented by the height of the right triangle.
Since you know the length of the hypotenuse (3960 miles plus 1450 feet) and you know the length of the base (3960 miles), you can readily find the height using the Pythagorean theorem (c^2 = a^2 + b^2).
I converted the distances from miles to feet. The numbers are large but manageable.
Earth's radius = 3960 miles = 20,908,800 feet.
Earth's radius plus height of the Sears Tower in Chicago = 20,908,800 + 1450 ft
= 20,910,250 feet.
c^2 = a^2 + b^2
(20,910,250)^2 = (20,908,800)^2 + h^2 Solve for h
h^2 = 4.37238555063X10^14 - 4.3717791744X10^14
h^2 = 60,637,622,500
h = 246,247 feet = 46.64 miles
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