Answer to Question #199163 in Trigonometry for Hetain

Question

From the top of an 80-ft. building, the angle of elevation of the top of a taller building is 49 and

the angle of depression of the base of this building is 62. Determine the height of the taller

building to the nearest foot.

62°

49°

80 ft.

a. 211 ft. b. 112 ft. c. 129 ft. d. 276 ft.

Answer 1

1) Firstly we should find the distance between two buildings. It can be done using the angle of depression, 62. We can imagine a triangle ABC formed by the distance between the buildings, AB = x, the height of the shorter building, BC = 80ft, which is projected to the taller building, and the distance between the bottom of the higher building and the top of the lower one, CA. The angle of depression ∠ CAB = 62, this triangle is right, so we can find the ∠ ACB to use it for the law of sines.

$∠ ACB = 180° - ∠ ABC - ∠ CAB = 180° - 90° - 62° = 28°$

The law of sines is an equation relating the lengths of the sides of a triangle to the sines of its angles.

$$a/sin A=b/sin B = c/Sin C=2R$$

It helps to find a side of a triangle with 2 angles and other side which are known.

For the present task:

$$sin(∠ACB)/BA=sin(∠CAB)/BC$$

$AB=sin(∠ACB)*BC/sin(∠CAB)= sin(28°)*80/sin(62°)=$

$$=0.469*80/0.883=42.5 (ft)$$

Then image another triangle, ABD, where D is the top of the higher building. Using the same law of sines and the angle of elevation, 49, we can make the following proportion:

$$sin(∠ADB)/BA=sin(∠DAB)/BD$$

$$BD=sin(∠DAB)*AB/sin(∠ADB)$$

$BD = sin(49)*42.5/sin(180-90-49)=0.755*42.5/0.656 = 48.72$

Adding two parts of the taller building, we have:

$$CD = CB + BD = 80 + 48 = 128.72 (ft)$$

Rounding to the nearest foot, we have $CD ≈ 129 ft.$

Answer: the height of the taller building is 129 ft, c.