A tree on a hillside casts a shadow 200 ft down the hill. If the angle of inclination of the hillside is 23 to the horizontal and the angle of elevation of the sun is 54, find the height of the tree. Please round the answer to the nearest integer.
draw a diagram you will see that
(h+200 sin 23°)/(200 cos 23°) = tan 54°
h = 175.25
To find the height of the tree, we can use the trigonometric concept of similar triangles. Let's break down the problem:
We have a tree on a hillside, and it casts a shadow down the hill. This creates a right triangle between the tree, its shadow, and the point where the shadow meets the horizontal ground.
The given information states that the angle of inclination of the hillside (angle between the hillside and the horizontal) is 23 degrees, and the angle of elevation of the sun (angle between the sun's rays and the horizontal) is 54 degrees.
To solve the problem, we need to determine the height of the tree, which is the opposite side of the triangle formed by the tree, its shadow, and the point where the shadow meets the horizontal ground.
Let's denote the height of the tree as x.
Now, we can use the trigonometric concept of tangent to relate the angles and sides of the triangle:
tangent(angle) = opposite / adjacent.
In this case, we have:
tangent(23°) = x / 200.
Now, let's solve for x:
x = 200 * tangent(23°).
Using a calculator, evaluate the tangent of 23 degrees:
tangent(23°) ≈ 0.4245.
Now, substitute this value back into the equation:
x = 200 * 0.4245 ≈ 84.9.
Rounding to the nearest integer, the height of the tree is approximately 85 feet.