QuestionA surveyor sees a building at $34^{\circ}$ elevation, 2544 ft away. Find distance to the building, building height, antenna angle, and height.

Studdy Solution

STEP 1

Assumptions1. The angle of elevation from the position of a surveyor on the ground to the top of the building is $34^{\circ}$ . . The distance from the surveyor's position to the top of the building is2544 feet.
3. The distance from the surveyor's position to the top of the antenna is2684 feet.
4. The antenna is on top of the building.
5. We are assuming a right triangle between the surveyor, the top of the building, and the base of the building. Another right triangle is formed between the surveyor, the top of the antenna, and the base of the building.

STEP 2

(a) To find the distance from the surveyor to the base of the building, we can use the tangent of the angle of elevation, which is opposite/adjacent. In this case, the opposite side is the height of the building (which is the distance from the surveyor to the top of the building), and the adjacent side is the distance from the surveyor to the base of the building, which we are trying to find.
$\tan(\theta) = \frac{opposite}{adjacent}$

STEP 3

Rearrange the equation to solve for the adjacent side (the distance from the surveyor to the base of the building).
$adjacent = \frac{opposite}{\tan(\theta)}$

STEP 4

Plug in the given values for the opposite side (the distance from the surveyor to the top of the building) and the angle of elevation.
$adjacent = \frac{2544}{\tan(34^{\circ})}$

STEP 5

Calculate the distance from the surveyor to the base of the building.
$adjacent \approx3750\, feet$

STEP 6

(b) To find the height of the building, we can use the cosine of the angle of elevation, which is adjacent/hypotenuse. In this case, the adjacent side is the distance from the surveyor to the base of the building, and the hypotenuse is the distance from the surveyor to the top of the building.
$\cos(\theta) = \frac{adjacent}{hypotenuse}$

STEP 7

Rearrange the equation to solve for the hypotenuse (the distance from the surveyor to the top of the building).
$hypotenuse = \frac{adjacent}{\cos(\theta)}$

STEP 8

Plug in the given values for the adjacent side (the distance from the surveyor to the base of the building) and the angle of elevation.
$hypotenuse = \frac{3750}{\cos(34^{\circ})}$

STEP 9

Calculate the height of the building.
$hypotenuse \approx2544\, feet$

STEP 10

(c) To find the angle of elevation from the surveyor to the top of the antenna, we can use the inverse tangent function, which gives the angle from the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the antenna (which is the distance from the surveyor to the top of the antenna), and the adjacent side is the distance from the surveyor to the base of the building.
$\theta = \tan^{-}\left(\frac{opposite}{adjacent}\right)$

STEP 11

Plug in the given values for the opposite side (the distance from the surveyor to the top of the antenna) and the adjacent side (the distance from the surveyor to the base of the building).
$\theta = \tan^{-}\left(\frac{2684}{3750}\right)$

STEP 12

Calculate the angle of elevation from the surveyor to the top of the antenna.
$\theta \approx35.5^{\circ}$

STEP 13

(d) To find the height of the antenna, subtract the height of the building from the distance from the surveyor to the top of the antenna.
$Height\, of\, antenna = Distance\, to\, top\, of\, antenna - Height\, of\, building$

STEP 14

Plug in the given values for the distance from the surveyor to the top of the antenna and the height of the building.
$Height\, of\, antenna =2684 -2544$

STEP 15

Calculate the height of the antenna.
$Height\, of\, antenna =140\, feet$ The surveyor is approximately3750 feet away from the base of the building. The building is approximately2544 feet tall. The angle of elevation from the surveyor to the top of the antenna is approximately $35.5^{\circ}$ . The antenna is approximately140 feet tall.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms