A 40-foot T.V. antenna stands on top of a building. From a point on the ground, the angle of elevation to the top and to the bottom of the antenna measures 56° and 42° respectively. How tall is the building?
Answers
Answer:
61.5
Step-by-step explanation:
In this case to calculate the height, we do the following:
The first is the graphic, attached image.
Then the calculations.
b / d = tan (42 °) = 0.9
d = b / 0.9
Then we have to:
(40 + b) / d = tan (56 °) = 1.48
(40 + b) = d * 1.48
Replacing we have:
(40 + b) = (b / 0.9) * 1.48
40 + b = 1.65 * b
1.65 * b - b = 40
b = 40 / 0.65
b = 61.5
Therefore the height is approximately 61.5
Final answer:
Using the tangent function of trigonometry for the two given angles, we can set up two equations. We solve these equations by substituting the fabricate distance from one into the other, providing the height of the building.
Explanation:
To find the height of the building, we'll use the tangent function of trigonometry. In this case, the tangent of an angle in a right triangle is defined as the opposite side divided by the adjacent side. Therefore, we create two equations using the two provided angles and the respective opposite sides (antenna and building + antenna), as we know the distances are proportional to the tangent of their angles when the adjacent side (distance from the point on the ground to the building) is the same.
- For the entire structure (building + antenna): tan(56 degrees) = (Height of building + 40 feet) / Distance from the building.
- For the antenna only: tan(42 degrees) = 40 feet / Distance from the building.
We can solve this system of equations by substitution. Since tan(42 degrees) = 40 feet / Distance from the building, it means the Distance from the building = 40 feet / tan(42 degrees). Plugging the Distance into equation 1 and solving for Height of building, we get: Height of building = tan(56 degrees) * Distance from building - 40 feet = tan(56 degrees) * (40 feet / tan(42 degrees)) - 40 feet. This gives us the height of the building.
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Proportions:
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To learn more about proportions, you can take a look at brainly.com/question/24372153