A passenger is an airplane at an altitude of a=20 kilometers sees two towns directly to the east of the plane. the angles of depression to the towns are 28 degrees and 55 degrees. how far apart are the towns
my answer is:
sin 55 = 20/y
y=20/sin55
y=24.42
55-28=27
sin27/x=sin28/24.42
24.42sin27/sin28=23.61
is my answer correct?
it's okay my answers are correct
I don't know who confirmed your answers. Your values for y give the line-of-sight distances, not distance on the ground.
The distance is actually
20 cot28° - 20 cot55°
To find the distance between the two towns, we can use trigonometry and the concept of angles of depression. Let's assume the distance between the two towns is 'x' kilometers.
From the given information, we can draw a triangle depicting the situation. The airplane is at the vertex of the triangle, and the two towns are at the base of the triangle. The angles of depression are the angles between the horizontal line and the line of sight from the airplane to each town.
We have two angles of depression: 28 degrees and 55 degrees. Since the angles are measured from the horizontal line, we can conclude that the sum of these two angles is 180 degrees (as they form a straight line).
To solve for the distance 'x', we will use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.
For the first town (angle of depression = 28 degrees):
Tangent(28 degrees) = opposite side / adjacent side
Tangent(28 degrees) = a / x
For the second town (angle of depression = 55 degrees):
Tangent(55 degrees) = opposite side / adjacent side
Tangent(55 degrees) = a / (x + d)
Since the towns are directly to the east of the airplane, their distances to the airplane on the horizontal line will be the same. We will call this distance 'd'.
Rearranging the first equation to solve for 'a', we get:
a = x * Tangent(28 degrees)
Substituting this value into the second equation, we get:
Tangent(55 degrees) = (x * Tangent(28 degrees)) / (x + d)
Now, we have an equation with two variables, 'x' and 'd'. To solve for these variables, we need an additional equation. Suppose we know the height of the airplane, 'a', and can assume a certain value for it, we can substitute it into the equation and solve for 'x' and 'd'.
Let's assume the height of the airplane is 20 kilometers. We can substitute this value into the equation as follows:
Tangent(55 degrees) = (x * Tangent(28 degrees)) / (x + d)
Tangent(55 degrees) = (x * Tangent(28 degrees)) / (x + d)
Tangent(55 degrees) = (x * Tangent(28 degrees)) / (x + d)
Tangent(55 degrees) * (x + d) = (x * Tangent(28 degrees))
(x + d) * Tangent(55 degrees) = x * Tangent(28 degrees)
(x * Tangent(55 degrees)) + (d * Tangent(55 degrees)) = (x * Tangent(28 degrees))
x * (Tangent(55 degrees) - Tangent(28 degrees)) = d * Tangent(55 degrees)
x = (d * Tangent(55 degrees)) / (Tangent(55 degrees) - Tangent(28 degrees))
Now we have an equation where 'x' is expressed in terms of 'd'. By plugging in different values for 'd' and using a scientific calculator to evaluate the tangent functions, we can find the corresponding values of 'x'. This will ultimately give us the distance between the two towns.