When sitting atop a tree and looking down at his pal joey, the angle of depression of mack’s line of sight is 53°51. if a joey is known to be standing 25 feet from the base of the tree, how tall is the tree( to the nearest foot)? a 36 ft b 38ft c 34ft d 40ft

  C  34ft

Step-by-step explanation:

SOH CAH TOA reminds you that ...

  Tan = Opposite/Adjacent

In this geometry, the height of the tree is the side Opposite the angle of elevation, with the Adjacent side being the one that is the 25 ft distance from Joey to the tree. The angles of elevation and depression are equal.

Hence ...

  tan(53°51') = height/(25 ft)

  height = (25 ft)·tan(53°51') ≈ 34.22 ft ≈ 34 ft.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!