Problema Solution
A tree is planted at a point (M) on horizontal ground.Two points A and B on the ground are 100 feet apart. The angles of elevation of the top of the tree (T) from the points A and B are 45 DEGREE and 30 DEGREE respectively. The measure of Triangle AMB is 60 DEGREE. Find the height of the tree
Answer provided by our tutors
let
h = the height of the tree
make good drawing and notice the right triangles. we will use tangent trigonometry function:
tan 45 = h/AM
h/AM = 1 (tan 45 = 1)
h = AM
tan 30 = h/BM
h/BM = pi/6 (tan 30 = pi/6)
since h = AM we have AM/BM = pi/6
using the Law of cosines we have:
100^2 = AM^2 + BM^2 - 2 AM BM cos 60
AM^2 + BM^2 - 2 AM BM (1/2) = 100^2 (cos 60 = 1/2)
AM^2 + BM^2 - AM BM = 10000
lets denote AM = a, BM = b
now we have the system of equations:
a/b = pi/6 => a = pi*b/6 plug into
a^2 + b^2 - a*b = 10000
(pi*b/6)^2 + b^2 - (pi*b/6)*b = 10000
pi = 3.14
(3.14*b/6)^2 + b^2 - (3.14*b/6)*b = 10000
b = 115.43
a = pi*b/6
a = 3.14*115.43/6
a = 60.41 ft
since h = AM = a we have our answer
the tree is 60.41 feet high!