A tree grows vertically on a slope which has an incline of 27 degrees. When the sun is at an elevation of 78 degrees, the tree casts a shadow of 48 feet directly down the slope.
sketch a line making an angle of 27° with the horizontal.
Label the length of the tree's shadow on that line as AB, with B the foot of the tree.
Let the top of the tree be C
(so we need BC)
from A draw a horizontal and let CB extended hit that horizontal at D
in the right-angle triangle , angle BAD = 27° so
angle ABD = 63°, making angle ABC = 117°
So in triangle ABC,
AB = 48
angle ABC = 117°,
angle CAB = 78-27 = 51°
making angle ACB = 12°
by the sine law:
BC/sin51 = 48/sin12
BC = 48sin51/sin12
carry on
To determine the height of the tree, we will use the concept of trigonometry. In this scenario, we have a slope with an incline angle of 27 degrees and the sun at an elevation of 78 degrees. The tree casts a shadow of 48 feet down the slope.
To begin, we will use the concept of similar triangles. The tree, its shadow, and the slope form a right triangle. The angle of elevation of the sun forms another right triangle with the slope. Since the two triangles are similar, we can set up a proportion to find the height of the tree.
Let's define the variables:
- h: the height of the tree
- x: the unknown length that we need to find (the adjacent side of the angle of 78 degrees, which is parallel to the tree's shadow)
- y: the length of the tree's shadow (48 feet)
In the triangle formed by the slope, the tree's height, and its shadow, we have the following trigonometric ratio:
tan(27 degrees) = h / y
To solve for h, we rearrange the equation:
h = y * tan(27 degrees)
Now, we need to find x, which is parallel to the tree's shadow and adjacent to the angle of 78 degrees. We can use the triangle formed by the sun's elevation angle, the slope, and x. In this triangle, we have the following trigonometric ratio:
tan(78 degrees) = h / x
Rearranging the equation, we find:
x = h / tan(78 degrees)
Now, we substitute the value of h from the first equation into the second equation:
x = (y * tan(27 degrees)) / tan(78 degrees)
Finally, we can substitute the given values into the equation to calculate the height of the tree:
x = (48 * tan(27 degrees)) / tan(78 degrees) ≈ 23.594 feet
Therefore, the tree is approximately 23.594 feet tall.