You are standing 40 meters from the base of the tree that is leaning 8° from the vertical away from you, the angle of elevation from your feet to the top of the tree is 20°50, what is the height of the tree?
The tree is about 16.28 meters tall.
First, we draw a diagram to represent the problem.
To convert the angle measurement from minutes to a decimal, divide the minutes by 60, and add that to the degrees. That’s our new angle measurement.
##50/60= 0.833+20= 20.833##
To find the angle the tree makes with the ground, first subtract 8 from 90. You can see that the vertical dotted line is parallel with the ground. From this you get 82º. The tree is 82º from the ground on the left, but we need the angle between the tree and the ground on the right. The two angles are complementary, so subtract 82 from 180.
The angle the tree makes with the ground in our triangle is 98º.
You can find the third angle because all the angles of a triangle add up to 180.
##180-98-20.883=61.117##
Fill in your diagram with this new information like so:
We now have one side and all three angles, and we can use to find the height of the tree.
Law of Sines – ##a/sinA=b/sinB=c/sinC##
The lowercase letters represent sides, and the uppercase their corresponding angles. Plug the information into the formula.
##40/sin(61.117º) = x/sin(20.883º)##
Solve the equation with a calculator and round to two decimal places.
##x= 16.28##
The tree is about 16.28 meters tall.
