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 [tex]\\ tan(\alpha)=\frac{opposite}{adjacent}[/tex], 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 [tex]\\tan(11)=\frac{H}{1mi}[/tex], and that [tex]\\ H=tan(11)*1mi=0.1944*1mi=0.1944mi[/tex], 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:
[tex]\\ \frac{5280ft}{1mi}=\frac{X}{0.1944mi}[/tex] ⇒
[tex]\\ X =\frac{5280ft*0.1944mi}{1mi}[/tex], 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°.
