Question

The angle of elevation to the top of the Empire State Building in New York is found to be 11 degrees from the ground at a distance of 1 mi from the base of the building. Using this information, find the height of the Empire State Building.

Answer 1

Answer:

The height of the Empire State Building is 0.1944miles (1026.43ft)

Step-by-step explanation:

This question can be answered using Trigonometry, and we can draw a sketch from the information provided. There is one attached.

From the problem's statement, we know that the angle of elevation to the top of the building from the ground is α=11°, and there is a distance of 1mi from the base of the building.

As we can see from the sketch, the given information permits us to draw a right-angled triangle and we can find the height H using trigonometric functions.

We need to remember that in a right-angled triangle, tangent function is defined as $\\ tan(\alpha)=\frac{opposite}{adjacent}$, that is, the ratio of the opposite side to the angle in question (in this case α) to the adjacent side to this angle.

The opposite side of angle α is H, and the adjacent side is the distance given, that is, 1 mile.

Looking at the sketch attached, we can see that $\\tan(11)=\frac{H}{1mi}$, and that $\\ H=tan(11)*1mi=0.1944*1mi=0.1944mi$, so the height of the Empire State Building, according to this information, is H = 0.1944mi.

The value of the tangent of the angle α was rounded to tan(11°)=0.1944.

A value of tan(11°)=0.19438030913771848424...could be found in WolframAlpha's website.

To find the equivalent distance in feet, we know that there are 5280ft in a mile, so, using proportions:

$\\ \frac{5280ft}{1mi}=\frac{X}{0.1944mi}$ ⇒

$\\ X =\frac{5280ft*0.1944mi}{1mi}$, which results in 1026.43ft.

This problem could be solved also using the Law of Sines, but using more steps, and knowing that the sum of the internal angles of a rigth-angled triangle equals 180°.

Answer 2

Draw a picture of a building and a one mile (or 5280 feet) distance to a point from the base of the building. 

The angle of elevation is given as 11° and we want the height of the building or x. 

tan(11°) = x/5280 

5280 tan(11°) = x 

1026.3 feet = x 


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