From the top of a 40m fire tower, a fire ranger observes smoke in tower
locations. One has a an angle of depression of 10o, and the
other has an angle of depression of 7o. Calculate the
distance between the smoke sightings when they are
a.) on opposite sides of the tower and in line with the tower.
b.) in perpendicular directions from the tower.
a) The distance to each fire is
X = 40/tan A where A is the depression angle. That would be 325.8 m for one fiore and 226.9 m for the other. If they are in opposite directions along the same line, add the two for their distance apart.
b) If the directions to the two fires are at right angles as seen from the tower, calculate the hypotenuse of the triangle (fire1, fire2 and tower) to get the distance between the fires.
thank you!
To find the distance between the smoke sightings, we can use trigonometry.
a.) For the scenario where the smoke sightings are on opposite sides of the tower and in line with the tower, we can use the concept of angles of depression to form a right triangle.
Let's label the distance between the smoke sightings as 'd'.
In the right triangle, the height of the tower is the opposite side, and the distance 'd' is the adjacent side. The angle of depression of 10 degrees is the angle between the adjacent side and the hypotenuse.
Using the trigonometric function tangent, we can set up the equation:
tan(10 degrees) = opposite side / adjacent side
Since we know that the height of the tower is 40m, the opposite side is 40m.
tan(10 degrees) = 40 / d
To solve for 'd', we rearrange the equation as:
d = 40 / tan(10 degrees)
Using a calculator, we can find that tan(10 degrees) ≈ 0.1763.
Substitute this value into the equation:
d = 40 / 0.1763
Calculating this gives us:
d ≈ 226.63 meters
Therefore, the distance between the smoke sightings is approximately 226.63 meters.
b.) For the scenario where the smoke sightings are in perpendicular directions from the tower, we can use the concept of similar triangles.
Let's label the distance between the base of the tower and one of the smoke sightings as 'a', and the distance between the base of the tower and the other smoke sighting as 'b'.
We have two right triangles: one formed by the fire tower, the base of the tower, and one smoke sighting, and another formed by the fire tower, the base of the tower, and the other smoke sighting.
Both these triangles have the same angle of depression of 7 degrees, and we can consider them as similar triangles.
From the similar triangles, we can set up the proportion:
a / b = tan(7 degrees)
Since we know the height of the tower is 40m, we can set up the following relationship:
a + b = 40
Solving these two equations simultaneously will give us the values of 'a' and 'b'.
Rearrange the first equation:
a = b * tan(7 degrees)
Substitute this into the second equation:
b * tan(7 degrees) + b = 40
Simplifying:
b * (tan(7 degrees) + 1) = 40
b = 40 / (tan(7 degrees) + 1)
Using a calculator, we find that tan(7 degrees) ≈ 0.1228.
Substitute this value into the equation:
b ≈ 40 / (0.1228 + 1)
Calculating this gives us:
b ≈ 282.13 meters
Since 'a + b = 40', we can find 'a':
a ≈ 40 - b
a ≈ 40 - 282.13
a ≈ -242.13 meters
However, since distance cannot be negative, we can ignore this negative value.
Therefore, the distance between the smoke sightings in perpendicular directions from the tower is approximately 282.13 meters.