1. ## 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.

2. The Sear's tower is 1450/5280 miles tall.

You have a triangle. Use Pythagoras.

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

$\displaystyle \sqrt{(\frac{2091025}{528})^{2}-3960^{2}}=\frac{455\cdot\sqrt{2929}}{528}\approx{4 6.64} \;\ miles$

3. ## 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?

4. Hello, symmetry!

You need a better diagram . . .

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

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

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

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

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

5. 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

6. ## 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!