bearing of a tower are taken from two points A and B 250m apart, along a straight line north-south path, A being due north of B.the bearing of the tower from A to B are 140* and 110* respectively.choose two option that gives a distance from A and B of the tower to nearest meter.
a)231 b)321 c)355 d)405 e)453 f)470 g)504 h)554
To find the distance from each point (A and B) to the tower, we can use trigonometry with the given bearings.
Let's consider point A first. We know the bearing from point A to the tower is 140°. Since A is due north of B, we can draw a right triangle with the tower as the vertex, the line from the tower to point A as the hypotenuse, and the north-south path as the base.
In this triangle, the angle opposite the base is 90°. The angle between the north-south path and the line to the tower is the bearing, which is 140°. So the angle between the line to the tower and the base is 90° - 140° = -50°.
We can use the tangent function to find the ratio between the height of the tower and the distance from point A to the tower:
tan(-50°) = height / distance_from_A_to_tower.
Rearranging the equation, we have:
height = tan(-50°) * distance_from_A_to_tower.
Similarly, we can do the same for point B. The bearing from point B to the tower is 110°. The angle between the line to the tower and the base in the triangle is 90° - 110° = -20°.
Again, using the tangent function, we can find the ratio:
tan(-20°) = height / distance_from_B_to_tower.
Rearranging the equation, we have:
height = tan(-20°) * distance_from_B_to_tower.
Now, we know that the distance between points A and B is 250m. So we can express the distance from point A to the tower as:
distance_from_A_to_tower = 250 - distance_from_B_to_tower.
We can substitute this value into the equation for height in terms of point A:
height = tan(-50°) * (250 - distance_from_B_to_tower).
Similarly, we can express the distance from point B to the tower in terms of point B:
distance_from_B_to_tower = 250 - distance_from_A_to_tower.
And substitute this value into the equation for height in terms of point B:
height = tan(-20°) * (250 - distance_from_A_to_tower).
Now, we need to find the values of distance_from_A_to_tower and distance_from_B_to_tower that give us a height to the nearest meter.
To do this, we can iterate through the answer options and substitute them into our equations to see if the resulting height is nearest to an integer value.
Let's start with option a)231.
For point A:
distance_from_A_to_tower = 231m
distance_from_B_to_tower = 250m - 231m = 19m
For point B:
distance_from_B_to_tower = 231m
distance_from_A_to_tower = 250m - 231m = 19m
Now, substitute these values back into the equations for height:
height_A = tan(-50°) * (250m - 19m)
height_B = tan(-20°) * (250m - 19m)
Calculate the heights and compare them to the nearest meter. Repeat this process for the other answer options. The option(s) that give a height to the nearest meter will be the correct answer(s).