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| = 57 m,
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.
Hope you made a sketch
On mine I labelled the top of the building P and its bottom Q
In triangle ABP, angle ABP = 143 , so angle APB = 12°
by the sine law:
BP/sin25 = 57/sin12
BP = 57sin25/sin12 = ....
repeat for AP
The explanation is not clear enougj
To solve this problem, we can use trigonometry and the property that the sum of the angles in a triangle is 180 degrees.
Let's start by drawing a diagram to visualize the problem:
B B
│ │
│ │
│ │
└─────────┴─────────┘
A A
Let's label the distance from A to the top of the building as x, and the distance from B to the top of the building as y.
Now, we can use the tangent function to relate the angles of elevation to the distances x and y:
tan(25°) = height of the building / x ...(1)
tan(37°) = height of the building / y ...(2)
We can rearrange equation (1) to solve for the height of the building:
height of the building = x * tan(25°)
Similarly, from equation (2):
height of the building = y * tan(37°)
Since the height of the building is the same in both equations, we can set these two expressions equal to each other:
x * tan(25°) = y * tan(37°)
Now, let's solve for x in terms of y:
x = (y * tan(37°)) / tan(25°)
We also know that the distance between A and B is given as 57 m. Therefore, we can write:
x + y = 57
Now, we have a system of equations:
x = (y * tan(37°)) / tan(25°)
x + y = 57
We can now solve this system of equations.
Using the first equation, we can express x in terms of y:
x = (y * 0.7536) / 0.4663
Substituting this into the second equation:
(y * 0.7536) / 0.4663 + y = 57
Multiplying both sides by 0.4663 to eliminate the denominator:
0.7536y + 0.4663y = 57 * 0.4663
1.2199y = 26.6391
y = 26.6391 / 1.2199
y ≈ 21.80 m
Now, we can substitute this value of y back into the equation for x:
x = (21.80 * 0.7536) / 0.4663
x ≈ 35.25 m
Therefore, the distance from point A to the top of the building is approximately 35.25 meters, and the distance from point B to the top of the building is approximately 21.80 meters.
To solve this problem, we can use the trigonometric concept of tangent.
Let's start by drawing a diagram to visualize the situation. We have a building with a top point, and from two different points A and B on level ground, we measure the angles of elevation to the top of the building. The distance between points A and B is given as 57 meters.
Now, let's denote the distance from point A to the top of the building as x meters and the distance from point B to the top of the building as y meters.
Using the tangent function, we can establish the following relationships:
tan(25°) = height of building (opposite side) / distance from A to the building (adjacent side)
tan(37°) = height of building (opposite side) / distance from B to the building (adjacent side)
Now, we can set up the equations:
tan(25°) = height of building / x
tan(37°) = height of building / y
To find the height of the building, we can isolate it in each equation:
height of building = tan(25°) * x
height of building = tan(37°) * y
Since the height of the building is the same in both equations, we can equate them:
tan(25°) * x = tan(37°) * y
Now, substituting the given values, we have:
0.4663 * x = 0.7536 * y
To obtain the values of x and y, we need another equation. We can use the fact that the distances from A and B add up to the given distance of 57 meters:
x + y = 57
Now, we have a system of two equations:
0.4663 * x = 0.7536 * y
x + y = 57
We can solve this system of equations to find the values of x and y.
By substituting the second equation into the first equation, we have:
0.4663 * (57 - y) = 0.7536 * y
Simplifying this equation, we get:
26.6 - 0.4663y = 0.7536y
Combining like terms:
0.4663y + 0.7536y = 26.6
Simplifying further:
1.2199y = 26.6
Dividing both sides by 1.2199, we find:
y ≈ 21.795
Substituting this value back into the equation x + y = 57:
x ≈ 35.205
Therefore, the distance from point A to the top of the building is approximately 35 meters, while the distance from point B to the top of the building is approximately 22 meters.