Answer:
a. distance of the surveyor to the base of the building = 2051.90 ft
b. height of the building = 1384 ft
c. Angle of elevation from the surveyor to the top of the antenna = 38.31°
d. Height of antenna = 237.08 ft
Step-by-step explanation:
The picture above is a illustration of the described event.
a = the height of the flag
b = the height of the building
c = distance of the surveyor from the base of the building
the angle of elevation from the position of the surveyor on the ground to the top of the building = 34°
distance from her position to the top of the building = 2475 ft
distance from her position to the top of the flag = 2615 ft
(a) How far away from the base of the building is the surveyor located?
using the SOHCAHTOA principle
cos 34° = c/2475
c = 0.8290375726 × 2475
c = 2051.8679921
c = 2051.90 ft
(b) How tall is the building
The height of the building = b
sin 34° = opposite /hypotenuse
0.5591929035 = b/2475
b = 0.5591929035 × 2475
b = 1384.0024361
b = 1384.00 ft
(c) What is the angle of elevation from the surveyor to the top of the antenna?
let the angle = ∅
cos ∅ = adjacent/hypotenuse
cos ∅ = 2051.90/2615
cos ∅ = 0.784665392
∅ = cos-1 0.784665392
∅ = 38.310258303
∅ = 38.31°
(d) How tall is the antenna?
height of the antenna = a
sin 38.31° = opposite/hypotenuse
sin 38.31° = (a + b)/2615
sin 38.31° × 2615 = (a + b)
(a + b) = 0.6199159917 × 2615
(a + b) = 1621.0803182
(a + b) = 1621. 08 ft
Height of antenna = 1621. 08 - 1384.00 = 237.08031822 ft
Height of antenna = 237.08 ft
A) Surveyor's distance from the base of the building; z ≈ 2052 ft
B) The height of the building is gotten as; y ≈ 1384 ft
C) The angle of elevation from the surveyor to the top of the antenna is;
∠R = 38.28
D) The height of the antenna is; x = 236 ft
I have attached an image below showing the triangle formed by this question and from the diagram, we see that;
x = height of the flag
y = height of the building
z = Surveyor's distance from the base of the building
Angle of elevation from the position of the surveyor on the ground to the top of the building = 34°
Distance from surveyor's position to the top of the building = 2475 ft
Distance from surveyor's position to the top of the antenna = 2615 ft
A) Using trigonometric ratios on the diagram, we can find z which is the Surveyor's distance from the base of the building. Thus;
z = 2475 cos 34°
z = 2475 × 0.8290
z ≈ 2052 ft
B) Again using trigonometric ratios on the diagram we can find y which is the height of the building. Thus;
y = 2475 sin 34°
y = 2475 × 0.5592
y ≈ 1384 ft
C) From the diagram attached, we can see that the angle of elevation from the surveyor to the top of the antenna is ∠R.
Again, using trigonometric ratios, we have;
∠R = cos⁻¹(2052/2615)
∠R = cos⁻¹0.7850
∠R = 38.28°
D) Since we are looking for the height of the antenna which is x, let us make use of the triangle with the dimensions x, 2475 and 2615.
The angle R for this triangle will be; 38.28 - 34 = 4.28°
Using cosine rule, we have;
x² = 2475² + 2615² - 2(2475 × 2615) cos 4.28
x² = 55698.3382
x = √55698.3382
x = 236 ft
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