Your altitude is 40,000 feet and you detect a target at a depression angle of 6 degrees and a range of 20 miles. what is the altitude of the target? Draw a picture.
first you would want to convert feet to miles or vice versa. Then, youll want to build the triangle because you start at 40,000 feet and then that goes at a six degree angle along the x axis for 20 miles. So you will want to put this on a cartesian plane because from there you will want to determine whether you need to use the law of cosines or the law of sines.
ignoring the curvature of the earth, I made a sketch and had the altitude of the target as x feet,
then
sin 6° = (40000-x)/105600
105600 sin6° = 40000 - x
x = 40000 - 105600sin 6° = appr. 28962 feet
To determine the altitude of the target, we need to use trigonometry. Let's first draw a diagram to visualize the scenario:
```
/|
/ |
/ | target
/___|
altitude (40,000 ft)
```
Here's how you can calculate the altitude of the target:
Step 1: Convert the given depression angle from degrees to radians.
The depression angle is given as 6 degrees. To convert it to radians, multiply it by π/180.
6° * (π/180) ≈ 0.105 radians (rounded to three decimal places)
Step 2: Use the tangent function to find the altitude of the target.
The tangent function relates the opposite side (altitude of the target) to the adjacent side (range to the target). In this case, the adjacent side is the range (20 miles).
Let's say the altitude of the target is 'x'. The tangent of the depression angle is equal to 'x' divided by the range.
tan(0.105) = x / 20
Rearranging the equation, we get:
x = tan(0.105) * 20
Using a scientific calculator, evaluate the tangent of 0.105, multiply it by 20, and you'll find the altitude of the target.