One of the tallest buildings in a country is topped by a high antenna. The angle of elevation from the position of a surveyor on the ground to the top of the building is 34°, the distance from her position to the top of the building is 2475 feet, and the distance from her position to the top of the antenna is 2615 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?

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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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

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step-by-step explanation: