answer this question from points A and B on level ground the angles of elevation of the top of a building are 25° and 37° respectively. if |AB|=57m,calculate, to the nearest metre, the distances of the top of the building from A and B if they are both on the same side of the building
If you draw the diagram and review your basic trig functions, then if the height is h, it becomes clear that
h cot25° - h cot37° = 57
h = 57/(cot25° - cot37°) = 69.73m
Tan25 = h/x
h =x*Tan25.
Tan37 = h/ (x-57)
h = (x-57)Tan37.
h = x*Tan25 = (x-57)Tan37
x*Tan25 = (x-57)Tan37
x*Tan25 = x*Tan37-57*Tan37
0.467x = 0.754x-43.
X = 150 m. = hor. distance from point A to foot of bldg.
x-57 = 150-57 = 93 m. = hor. distance from point B to foot of bldg.
h = x*Tan25 = 150*Tan25 = 70 m. = ht. of bldg.
sin25 = h/r = 70/r
r = 166 m. = distance from point A to top of bldg.
sin37 = h/r = 70/r
r = 116 m. = distance from point B to top of bldg.
To find the distances of the top of the building from points A and B, we can use trigonometry. Let's break down the problem step by step.
1. Draw a diagram: Start by drawing a diagram with the building, points A and B, and the angles of elevation. Label the angles as 25° and 37°, and the distance |AB| as 57m.
2. Identify right triangles: Notice that from each point (A and B), you can form a right triangle with the top of the building as the vertical side, and the distance from point A or B to the building as the horizontal side.
3. Use trigonometric ratios: We can use the tangent ratio to solve for the distances. The tangent of an angle is equal to the opposite side divided by the adjacent side.
For point A:
tangent(25°) = opposite side / adjacent side
tangent(25°) = height of the building / distance from A to the building
For point B:
tangent(37°) = opposite side / adjacent side
tangent(37°) = height of the building / distance from B to the building
4. Solve the equations: Rearranging the equations, we can solve for the height of the building and the distances from points A and B.
height of the building = (tangent(25°)) * (distance from A to the building)
height of the building = (tangent(37°)) * (distance from B to the building)
5. Substitute the values and calculate: Plug in the known values into the equations and calculate the results.
height of the building = tan(25°) * 57
distance from A to the building = height of the building / tan(25°)
height of the building = tan(37°) * 57
distance from B to the building = height of the building / tan(37°)
Using a scientific calculator, calculate the values:
height of the building ≈ 22.76 m (rounded to 2 decimal places)
distance from A to the building ≈ 57 / tan(25°) ≈ 60.97 m (rounded to the nearest metre)
distance from B to the building ≈ 57 / tan(37°) ≈ 57.84 m (rounded to the nearest metre)
Therefore, the distance of the top of the building from point A is approximately 61 meters, and the distance from point B is approximately 58 meters, both rounded to the nearest meter.