Chapter 8.1, Problem 66AYU

Willis Tower Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurement:

1. The angle of elevation from his position to the top of the building is $34°$.

2. The distance from his position to the top of the building is 2593 feet.

3. The distance from his position to the top of the antenna is 2743 feet.

(a) How far away from the (base of the) building is the surveyor located?

(b) How tall is the building?

(c) What is the angle of elevation from the surveyor to the top of the antenna?

(d) How tall is the antenna?

Source: Council on Tall Buildings and Urban Habitat

To determine

To find:

a. How far away from the base of the building is the surveyor located?

Answer to Problem 66AYU

a. The surveyor is located approximately 2150 feet from the building.

Explanation of Solution

Given:

Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurement:

1. The angle of elevation from his position to the top of the building is $34°$.

2. The distance from his position to the top of the building is 2593 feet.

3. The distance from his position to the top of the antenna is 2743 feet.

Formula used:

$\text{cos}\theta =\text{ adj}/\text{hyp}$

$c²=a²+b²$

Calculation:

a. Let $h$ be the height of the building.

Let be the distance between surveyor and the base of the building.

As per the surveyor the angle of elevation is $34°$.

$\text{cos}\theta =\text{ adj}/\text{hyp}$

$\cos 34°=\frac{x}{2593}$

$x=2593*\cos 34°$

$x\approx 2150$

The surveyor is located approximately 2150 feet from the building.

To determine

To find:

b. How tall is the building?

Answer to Problem 66AYU

b. The building is 1450 feet tall.

Explanation of Solution

Given:

Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurement:

1. The angle of elevation from his position to the top of the building is $34°$.

2. The distance from his position to the top of the building is 2593 feet.

3. The distance from his position to the top of the antenna is 2743 feet.

Formula used:

$\text{cos}\theta =\text{ adj}/\text{hyp}$

$c²=a²+b²$

Calculation:

b. To find $h$.

Using Pythagoras theorem,

$h²=2593²-2150²$

$h\approx 1450$

The building is 1450 feet tall.

To determine

To find:

c. What is the angle of elevation from the surveyor to the top of the antenna?

Answer to Problem 66AYU

c. Angle of elevation from the surveyor to the top of the antenna is $38.4°$.

Explanation of Solution

Given:

Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurement:

1. The angle of elevation from his position to the top of the building is $34°$.

2. The distance from his position to the top of the building is 2593 feet.

3. The distance from his position to the top of the antenna is 2743 feet.

Formula used:

$\text{cos}\theta =\text{ adj}/\text{hyp}$

$c²=a²+b²$

Calculation:

c. Let $\theta$ be the angle of inclination from the surveyor to the top of the antenna.

$\cos \theta =\frac{2150}{2743}$

$\theta ={\cos }^{-1}\frac{2150}{2743}$

$\theta =38.4°$

Angle of elevation from the surveyor to the top of the antenna is $38.4°$.

To determine

To find:

d. How tall is the antenna?

Answer to Problem 66AYU

d. The antenna is 253 feet tall.

Explanation of Solution

Given:

Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurement:

1. The angle of elevation from his position to the top of the building is $34°$.

2. The distance from his position to the top of the building is 2593 feet.

3. The distance from his position to the top of the antenna is 2743 feet.

Formula used:

$\text{cos}\theta =\text{ adj}/\text{hyp}$

$c²=a²+b²$

Calculation:

d. To find the height of the antenna, we need to find $a$.

${\left(h+a\right)}^2={2743}^2-x²$

${\left(h+a\right)}^2={2743}^2-{2150}^2$

$\left(h+a\right)=1703.39$ feet.

$h\approx 1450$ feet.

$a\approx 1703-1450\approx 253.4\text{ feet }\approx 253\text{ feet}$.

The antenna is 253 feet tall.