A tree on a hillside casts a shadow c = 210 ft down the hill. If the angle of inclination of the hillside is b = 21° to the horizontal and the angle of elevation of the sun is a = 51°, find the height of the tree. (Round your answer to the nearest foot.)
________ft
Draw a diagram.
T = top of tree
S = tip of shadow
B = base of tree
Draw a line L horizontally from the tip of the shadow.
Drop a vertical from the tree base to intersect line L at P.
Let
x = SP
y = PB
h = BT (height of tree)
y = 210 sin21°
x^2+y^2 = 210^2
(y+h)/x = tan51°
I get h=166.847
To find the height of the tree, we can use trigonometry.
We know that the shadow cast by the tree down the hill is 210 ft. This is the adjacent side of a right triangle formed by the hillside and the shadow.
The angle of inclination of the hillside (b) is given as 21°. This is the angle between the hillside and the horizontal ground.
The angle of elevation of the sun (a) is given as 51°. This is the angle between the horizontal ground and the line from the top of the tree to the sun.
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side of a right triangle.
In this case, the opposite side is the height of the tree (h), and the adjacent side is the length of the shadow (c).
Therefore, we have the equation:
tan(a) = h/c
Substituting the given values, we have:
tan(51°) = h/210 ft
Now, we can solve for h by multiplying both sides by 210 ft:
h = 210 ft * tan(51°)
Using a calculator, we find that tan(51°) ≈ 1.305.
Now, we can calculate the height of the tree:
h = 210 ft * 1.305
h ≈ 275.03 ft
Rounding to the nearest foot, the height of the tree is approximately 275 ft.
Therefore, the height of the tree is 275 ft.