Atop 50m tower, Jack sees smoke in two areas. One is on a bearing of 40degrees with an angle of depression of 8degrees, and the other on a bearing of 205degrees with and angle of depression of 13degrees. How far apart are the smoke sources?
I'm not sure how bearings work in this question.
I think we must assume that the tower is in relatively flat plane.
I don't know how well you can sketch 3-D diagrams, but mine has two right-angled triangles.
One showing the first fire, has a base angle of 8° and a height of 50, so the distance of the fire from is:
50/base = tan8, base = 50/tan8°
similarly the 2nd right angled triangle has a base angle of 13° and the distance of the fire from the tower is 50/tan13°
Now the bearing:
the 2nd is 204° and the 1st is 40°
so in now have a triangle with sides 50/tan8 and 50/tan13 with an angle of 204-40 or 165° between them
let the distance between the two fires be x
x^2 = (50/tan8)^2 + (50/tan13)^2 - 2(50/tan8)(50/tan13)cos 165°
I will let you do the button-pushing.
No worries! I'll explain how bearings work in this question and help you find the solution.
In the context of this question, a bearing is an angle measured clockwise from the north direction. It tells us the compass direction of an object or location with respect to a reference point. The reference point is usually the observer's position.
To solve this problem, we can use trigonometry, specifically the tangent function. The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Let's consider the first smoke source. According to the information given, it has a bearing of 40 degrees from the top of the tower. We can visualize this by drawing a line from the top of the tower at an angle of 40 degrees clockwise from the north direction.
The angle of depression is the angle formed between the line of sight from the top of the tower to the smoke source and the horizontal. In this case, the angle of depression is given as 8 degrees. This means that if we imagine a horizontal line from the top of the tower, the line of sight to the smoke source is 8 degrees below the horizontal.
Now, imagine drawing a right-angled triangle with one side being the height of the tower (50m). The opposite side of the angle of depression would represent the distance from the top of the tower to the smoke source.
Since we have the angle of depression and the length of the opposite side, we can use the tangent function to find the adjacent side (distance from the top of the tower to the smoke source).
Using the formula: tan(angle) = opposite side / adjacent side, we can rearrange the formula to find the adjacent side.
tan(8 degrees) = opposite side / adjacent side
To find the adjacent side, we can rearrange the formula:
adjacent side = opposite side / tan(8 degrees)
Plugging in the values, we have:
adjacent side = 50m / tan(8 degrees)
Now, you can use a scientific calculator or an online calculator to find the value of tan(8 degrees) and evaluate the expression to find the length of the adjacent side or the distance from the top of the tower to the first smoke source.
Repeat the same process for the second smoke source, which has a bearing of 205 degrees and an angle of depression of 13 degrees.
Once you have the distances from the top of the tower to both smoke sources, you can find the distance between the smoke sources by subtracting one distance from the other.
I hope this explanation helps! Let me know if you have any further questions.