Could I get some help with the word problem?

Question

After a severe storm, three sisters, April, May, and June, stood on their front porch and noticed that the tree in their front yard was leaning 3° from vertical toward the house. From the porch, which is 96 feet away from the base of the tree, they noticed that the angle of elevation to the top of the tree was 31°. Approximate the height of the tree. Round answer to two decimal places.

Answer 1

Draw a diagram, calculate the third angle, and then use the law of sines to determine the length of the tree. Then use the cosine function to calculate its new height after the storm.

Answer 2

Other tutors have covered the use of law of Sines/cosines. I want to show you that it is possible to calculate this from simple trigonometric logic.

Draw a triangle formed by the base of the tree (point A), the porch (point B) and the top of the tree (point C). Draw the angles in the triangle.

Given the tree is tilted at an angle 3º

Angle CAB, θ = (90-3º) = 87º

Angle CBA, φ = 31º (given)

Length AB = 96 ft (given)

Length AC = "h", actual height of the tree (to be estimated)

Drop a line segment perpendicular to the base AB from apex C of the triangle, meeting it at point D. Let

Length CD = p and AD = x

Now p is the common perpendicular shared between two right triangles ADC and BDC. From the triangles write the following expressions

tan θ = p/x

tan φ = p/(96-x)

Eliminate p and calculate x.

x = 2.927 ft

Now, from triangle ADC, calculate AC using the cosine function

Cos θ = x/h

h = 2.927/0.0523 = 55.97 ft.

Answer 3

180 - 34 = 146

sin34/96 = sin31/t

tsin34 = 90sin31

t = 90sin31/sin34

t = 97.72 ft