man on 12m tower, he observed two houses. the angle of depression to one of them was 6' and the other is 3'. how far apart the houses are. Calculate the distance between the house if:
a)the houses are on the same side of the tower
b)the houses are on opposite sides of the tower(in line with the tower)
c)the distance between the houses if one house is due south from the tower and the other is due east from the tower.
what were the answers to this question
To find the distances between the houses in the given scenarios, we will use trigonometry, specifically the tangent function.
Let's establish some notation:
- Let "x" represent the distance between the tower and the closer house.
- Let "y" represent the distance between the tower and the farther house.
a) When the houses are on the same side of the tower:
We can form a right-angled triangle with the tower as the vertical side and the line joining the two houses as the hypotenuse. In this case, we can use the tangent of the angle of depression to find the distance between the houses.
The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Since the angle of depression is the angle at the tower's top, the opposite side is the difference in height between the tower and the houses, which is 12m.
Therefore, using the tangent function:
tan(6') = 12/x
To solve for x, rearrange the equation:
x = 12/tan(6')
Similarly, for the distance y, we have:
y = 12/tan(3')
b) When the houses are on opposite sides of the tower (in line with the tower):
In this case, the angle of depression is formed at the feet of the observer and the line joining the two houses. We can set up a similar right-angled triangle with the distance between the houses as the opposite side and the height of the tower as the adjacent side.
Using the tangent function:
tan(6') = x/12
Rearranging the equation to solve for x:
x = 12*tan(6')
Similarly, for the distance y, we have:
y = 12*tan(3')
c) When one house is due south from the tower and the other is due east from the tower:
In this case, we have a right-angled triangle with the tower as the hypotenuse. The distance between the houses is represented by the two sides of the triangle.
Using the Pythagorean theorem:
y^2 + x^2 = 12^2
Since one house is south and the other is east, we have a right angle between the triangle's sides. Thus, the triangle is a 45-45-90 triangle. In this special triangle, the sides are in the ratio 1:1:√2.
So, x = y = 12/√2
Now, you can substitute the values of the angles or angle measurements into the equations to calculate the distances between the houses in each scenario.