a passenger in an airplane at an altitude of a = 20 kilometers sees two towns directly to the east of the plane. The angle of depression to the towns are 55˚ and 28˚. How far apart are the towns?
No idea what x and y are, you did not define them.
Since the altitude would be be measured perpendicular to the ground
label the position of the plane P and the bottom of the altitude as Q
Label the towns with angles of depression of 55˚ and 28 as A and B respectively.
tan 55 = 20/QA
QA = 20/tan55 = 14.004 km
tan28 = 20/QB
QB = 20/tan28° = 37.615 km
distance between towns = QB - QA = ....
It looks like you first found the hypotenuse of the smaller right-angled triangle
then used the sine law to find the distance between the towns
However, the way you presented the solution would not be acceptable.
1. no definitions of x and y
2. 55-28=27 <---- I had to guess what that calculation was supposed to be
3. from: sin27/x=sin28/24.42
to : 24.42sin27/sin28=23.61 you lost your variable
To find the distance between the two towns, we can use the concept of trigonometry. Let's consider the following diagram:
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a / | b
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Angle of depression A B
In this diagram, the airplane is at point A at an altitude of 20 kilometers. The two towns are at points B and C, and we need to find the distance BC.
First, let's find the distance AB using the angle of depression of 55˚. We can use the tangent function to calculate this distance. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is 20 kilometers (altitude) and the adjacent side is AB. So we have:
tan(55˚) = 20 / AB
To isolate AB, we rearrange the equation:
AB = 20 / tan(55˚)
Next, let's find the distance AC using the angle of depression of 28˚. Again, we can use the tangent function:
tan(28˚) = 20 / AC
Rearranging the equation for AC, we get:
AC = 20 / tan(28˚)
Now, the distance BC is equal to the difference between AB and AC:
BC = AB - AC
Substituting the values we calculated earlier, we have:
BC = (20 / tan(55˚)) - (20 / tan(28˚))
Calculating the values for AB, AC, and BC will give us the final answer for the distance between the two towns.
To find the distance between the towns, we can use trigonometry and apply the tangent function.
Let's denote:
- The distance between the towns as x.
- The height of the airplane above the ground as h.
From the given information, we know that:
- The angle of depression to the first town is 55˚.
- The angle of depression to the second town is 28˚.
- The height of the airplane is 20 kilometers.
Now, let's consider a right-angled triangle for each town, with the horizontal line connecting the towns as the hypotenuse.
For the first town:
We have:
- The opposite side is h.
- The adjacent side is x.
Using the tangent function, we can write:
tan(55˚) = h / x
For the second town:
We have:
- The opposite side is h.
- The adjacent side is x.
Again using the tangent function, we can write:
tan(28˚) = h / x
Since both triangles have the same height (h), we can set up the following equation:
tan(55˚) = tan(28˚) = h / x
Now, let's solve the equation to find the value of x:
tan(55˚) = tan(28˚)
h / x = h / x
We can see that the height (h) cancels out, so we are left with:
1 / x = 1 / x
This means that x can take any value, and the distance between the towns is not uniquely determined.
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?