Question

a tree grows vertically along a hillside,the hill is at 16 degrees angle to the horizontal, the tree cast an 18 meter shadow up the hill when the angle of the sun measures 68 degrees, what is the tree's height?

Answer 1

The height of the tree on the hillside can be determined by using the sine function with the given angle of the sun and length of the shadow to calculate the height.

Step-by-step explanation:

To solve for the height of the tree, we need to apply trigonometry. The hill makes a 16-degree angle with the horizontal, but the tree grows vertically, so it's perpendicular to the horizontal regardless of the hill's slope. We also know the shadow of the tree casts up the hill at a 68-degree angle from the horizontal. We use these angles to determine the right triangle with the tree height as one side, and the shadow as the hypotenuse.

To find the height (h) of the tree, we can use the sine function:

sin(68°) = h / shadow length

Since the shadow length is 18 meters, we rearrange the equation to solve for h:

h = 18 * sin(68°)

Calculating this gives us the height of the tree.

Answer 2

The idea is to create a triangle based upon the info given (see attached drawing). The bottom side (c) represents the length of the tree's shadow cast by the sun (18 ft), whose opposite angle (C) = 68°, the angle of the sun's elevation. The left side (b) is the height of the tree. The top side (a) is not needed, but it's opposite angle (A) is on order to find out B.
Since the hill incline is 16°, the angle A (angle tree makes with the hill) is 90-16 = 74°. Since C = 68 and A = 74, then B = 180-68-74, because all three angles must equal 180° within a triangle. So B = 38°.
Now that we have two angles and a side of our triangle, we can use the law of sines to calculate a missing side (b).
Law of sines: a/sinA = b/sinB = c/sinC
--> b = c×sinB / sinC
b = 18×sin38 / sin68
b = 11.082 / 0.927 = 11.95 ft