Question

A plane flies a straight course. on the ground directly below the flight path, observers 2 miles apart spot the plane at the same time. the plane's angle of elevation is 46° from one observation point and 71° from the other. how high is the plane?

Answer 1

The height of the plane is approximately 3.20 miles.

Step-by-step explanation:

To solve this problem, we can use trigonometry. Let's denote the height of the plane as h.

From the first observation point (where the angle of elevation is 46°), we can create a right triangle with the height of the plane as the opposite side and the distance between the two observation points (2 miles) as the adjacent side. Using the tangent function:

tan(46∘)= h/2

From the second observation point (where the angle of elevation is 71°), we can create another right triangle with the height of the plane as the opposite side and the same distance between the observation points (2 miles) as the adjacent side.

Using the tangent function:

tan(71∘)= h/2

Now, we have a system of two equations:

tan(46∘)= h/2

tan(71∘)= h/2

We can solve this system of equations to find the height (h) of the plane.

h=2⋅tan(46∘)

h=2⋅tan(71∘)

Now, let's calculate:

h=2⋅tan(46∘)

h≈2⋅1.035 (tan(46°) is approximately 1.035)

h≈2.07

h=2⋅tan(71∘)

h≈2⋅2.165 (tan(71°) is approximately 2.165)

h≈4.33

Now, we have two values for h from different observations. Since the height of the plane should be the same in both cases, we can take the average:

Average height= 2.07+4.33/2

Average height≈3.20

So, the height of the plane is approximately 3.20 miles.