HELP ME PLEASE ANY SMART PEOPLE

Question

The angle of depression from a plane to the runway is 24°. If the pilot's current altitude is 763m, how far is the plane from the runway?

Another question:

A tree makes a shadow of 31.5m. If the angle of elevation the sun makes with the ground is 47°, how tall is the tree?

Answer 1

To be able to do either problem, we need to identify how to do each problem. The best way to do this is to use right triangles.

Let's start off with the first problem. We know that the pilot's altitude is 763 m, and that it makes a 24-degree angle of depression to the runway. How do we find how far the plane is from the runway? (I'll do my best to create a triangle.)

Plane

| \

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763 m |24^o \

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x

We need to use SOH CAH TOA. In case you don't know what SOH CAH TOA is, it goes as follows:

1. SOH is "sin is opposite over hypotenuse", meaning that sin of an angle is equal to the opposite side divided by the hypotenuse.
2. CAH is "cos is adjacent over hypotenuse", meaning that cos of an angle is equal to the adjacent side divided by the hypotenuse.
3. TOA is "tan is opposite over adjacent", meaning that tan of an angle is equal to the opposite side divided by the adjacent side.

For this problem, we will be using TOA. We do as follows:

tan(θ) = x/y, where θ is the angle of depression, x is the distance the plane has till it reaches the runway, and y is its altitude. We now solve:

tan(24) = x/763

763tan(24) = x

339.7 m = x

Therefore, the distance between the plane and the runway is 339.7 meters.

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The second problem, we're going to be doing the same exact thing. Again, I'll attempt to do a triangle below:

| \

| \

| \

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x | 47^o \

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31.5 m

Again, we are going to be using tan to solve for the height of the tree.

tan(θ) = y/x, where θ is the angle the sun makes with the ground, x is the height of the tree, and y is the shadow of the tree. We now solve:

tan(47) = 31.5/x

xtan(47) = 31.5

x = 31.5/tan(47)

x = 29.4 m

Therefore, the height of the tree is 29.4 meters.

Answer 2

tan(θ) = opposite/adjacent so:

tan(24°) = 763/x or x = 763/tan(24°)

Plugging this into a calculator you get x = 1714 meters

Draw a picture of the second problem. You would be standing at the end of the shadow of the tree looking up at the angle or 47° to see the sun and the top of the tree at the same point. The tree height is "x", The distance from you to the base of the tree (aka, the length of the shadow) is 31.5 so you can write a similar tangent ratio as above.