Trigonometry real life scenario

Question

Nicole wants to find out the total height of her christmas tree. The star on the top is 0.24 meters tall. Nicole measured the angle of elevation( top of the star and tree) and depression (bottom of tree) using a clinometer and it measured to be 71° and 64° respectively. What is the total height of the tree? Round your answer to the nearest tenth.

Please insert a diagram if possible

Answer 1

Problem:

Nicole wants to find out the total height of her Christmas tree. The star on the top is 0.24 meters tall. Nicole measured the angle of elevation (top of the star and tree) and depression (bottom of tree) using a clinometer and it measured to be 71° and 64° respectively. What is the total height of the tree? Round your answer to the nearest tenth.

Solution:

Triangle:

a = hypotenuse

b = adjacent side

c = 0.24 m (small opposite side)

A = ?

B = ?

C = 71 - 64 = 7 degrees

Assign y as the height of the tree.

By the definition of sine = opp / hyp:

Sin 71 = (y + 0.24) / a

Sin 64 = y / b

Solving for a and b we get:

a = (y + 0.24) / Sin 71

b = y / Sin 64

c = 0.24

C = 7

Law of Cosines:

c^2 = a^2 + b^2 - 2ab*Cos C

Substitute a, b, c and C into this equation and you will have 1 equation with 1 unknown, y, or the height of the tree.

(c)^2 = (a)^2 + (b)^2 - 2(a)(b)*Cos C

(0.24)^2 = ((y + 0.24) / Sin 71)^2 + (y / Sin 64)^2 - 2((y + 0.24) / Sin 71)(y / Sin 64)*Cos 7

Multiply out, isolate the y^2, take its square root and you have the answer.