5. A backpacker notes that from a certain point on level ground, the angle of elevation to a point at the top of a tree is 34o. After walking 50 closer to the tree, the backpacker notes that the angle of elevation is 54o. Find the height of the tree.
To find the height of the tree, we can use trigonometry. Specifically, we can use the tangent function.
Let's assign some variables:
Let h be the height of the tree.
Let d be the distance between the backpacker and the tree at the starting point.
Let x be the distance the backpacker walks closer to the tree.
We have two right triangles. The first right triangle is formed by the backpacker, the top of the tree, and a point on the ground. The second right triangle is formed by the backpacker, the top of the tree (after walking closer), and a point on the ground.
In the first right triangle:
The angle of elevation to the top of the tree is 34 degrees.
The opposite side is the height of the tree, h.
The adjacent side is the distance between the backpacker and the tree at the starting point, d.
Using the tangent function, we can write:
tan(34) = h / d
In the second right triangle:
The angle of elevation to the top of the tree is 54 degrees.
The opposite side is still the height of the tree, h.
The adjacent side is the distance between the backpacker and the tree after walking closer, d - x.
Using the tangent function again, we can write:
tan(54) = h / (d - x)
Now we have two equations with two unknowns. We can solve this system of equations to find the height of the tree.
First, let's solve for d in terms of h in the first equation:
tan(34) = h / d
d = h / tan(34)
Now substitute this value of d in the second equation:
tan(54) = h / (d - x)
tan(54) = h / (h / tan(34) - x)
Now, we can solve this equation for x:
tan(54) = h / (h / tan(34) - x)
tan(54) = tan(34) / (1 - x / (h / tan(34)))
tan(54) = tan(34) / (1 - x * tan(34) / h)
Next, we can solve this equation for h:
h * tan(54) = tan(34) - x * tan(34)
h = (tan(34) - x * tan(34)) / tan(54)
Now we can substitute the value of h back into any of the original equations to solve for the height of the tree.