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.
someone plz reply, be very greatful
To solve this problem, we can use trigonometry, specifically the tangent function, which relates the angle of depression to the distance.
Let's first understand the situation:
A man is on top of a 12m tower, observing two houses. The angle of depression to the first house is 6' (6 minutes), and to the second house is 3' (3 minutes). We need to calculate the distance between the houses.
a) If the houses are on the same side of the tower:
To calculate the distance between the houses, we can imagine creating a right-angled triangle with the man on top of the tower, one of the houses, and the base of the tower forming the three sides of the triangle.
Let's call the distance between the tower and the first house "x" (in meters). Using the tangent function, we can say:
tan(6') = x/12
x = 12 * tan(6')
Similarly, let's call the distance between the tower and the second house "y" (in meters):
tan(3') = y/12
y = 12 * tan(3')
The distance between the houses is the sum of these two distances: x + y.
b) If the houses are on opposite sides of the tower (in line with the tower):
In this case, we create two right-angled triangles: one with the man on top of the tower, the first house, and the base of the tower; and the other with the man on top of the tower, the second house, and the base of the tower.
Using the same logic as before, we can calculate the distances between the tower and each house individually using the tangent function. Let's call the distance between the tower and the first house "x" (in meters) and the distance between the tower and the second house "y" (in meters):
tan(6') = x/12
x = 12 * tan(6')
tan(3') = y/12
y = 12 * tan(3')
The distance between the houses is the difference between these two distances: |x - y|.
c) If one house is due south from the tower and the other is due east from the tower:
In this case, we create a right-angled triangle with the man on top of the tower, the first house, and the base of the tower. From this triangle, we can find the distance between the tower and the first house using the tangent function.
Using the same logic as before, let's call the distance between the tower and the first house "x" (in meters):
tan(6') = x/12
x = 12 * tan(6')
Now, we can create another right-angled triangle with the man on top of the tower, the second house, and the base of the tower. This triangle will give us the distance between the tower and the second house using the tangent function.
Similarly, let's call the distance between the tower and the second house "y" (in meters):
tan(3') = y/12
y = 12 * tan(3')
To find the total distance between the houses, we can use the Pythagorean theorem, which states that, in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the distance between the houses.
The distance between the houses = √(x^2 + y^2).
By using these trigonometric calculations, we can determine the distances between the houses in the given scenarios.