# Chicago's Sears Tower

• Jan 10th 2007, 02:04 PM
symmetry
Chicago's Sears Tower
The tallest building in the world is the Sears Tower in the city of Chicago. If the observation tower is 1450 feet above the ground, how far can a person see standing in the observation tower using the fact Earth's radius is 3960 miles?

NOTE: 1 mile = 5280 feet.
• Jan 10th 2007, 02:42 PM
galactus
The Sear's tower is 1450/5280 miles tall.

You have a triangle. Use Pythagoras.

3960+(1450/5280)=3960+(145/528)=2091025/528

$\sqrt{(\frac{2091025}{528})^{2}-3960^{2}}=\frac{455\cdot\sqrt{2929}}{528}\approx{4 6.64} \;\ miles$
• Jan 10th 2007, 04:47 PM
symmetry
ok
In other words, we have a right triangle where one leg = 1450 and the other leg = 5280. I would need to find c = hypotenuse, right?
• Jan 10th 2007, 10:16 PM
Soroban
Hello, symmetry!

You need a better diagram . . .

Quote:

The tallest building in the world is the Sears Tower in the city of Chicago.
If the observation tower is 1450 feet above the ground,
how far can a person see standing in the observation tower
using the fact that Earth's radius is 3960 miles?

Code:

                *                 | \               h|  \                 |    \d               * * *    \         *      |      *       *        |      /*       *        R|    /  *                 |  / R     *          | /        *     *          *          *     *                    *       *                  *       *                *         *            *               * * *

The height of the tower is $h.$
The radius of the earth is $R.$
The distance from the top of the tower to the horizon is $d.$
$d$ and $R$ form a right angle.

From the right triangle: . $d^2 + R^2\:=\:(h + R)^2$
. . which simplifies to: . $d^2\:=\:2Rh + h^2\:=\:h(2R + h)$ [1]

We are told that: . $R = 3960$ and $h = \frac{1450}{5280} = \frac{145}{528}$

Substitute into [1]: . $d^2\:=\:\frac{145}{5287}\left(2\cdot3960 + \frac{145}{528}\right) \:=\:2175.075417$

Therefore: . $d \:\approx\:46.63$ miles.

• Jan 10th 2007, 10:34 PM
CaptainBlack
Quote:

Originally Posted by symmetry
In other words, we have a right triangle where one leg = 1450 and the other leg = 5280. I would need to find c = hypotenuse, right?

What you need here is a diagram, see below

RonL
• Jan 11th 2007, 03:35 AM
symmetry
ok
I want to thank everyone who took time out to answer this question, especially soroban and captainblack for their excellent diagrams of the Earth and triangle.

A job well-done!

I fully understand this question now.

Thanks!