A surveyor stands 150ft from the base of a tree and measures the angle of elevation to be 46.2. His eye level is 6feet above the ground. What is the height of the tree to the nearest foot?
I'd be interested in seeing your work. I get
(h-6)/150 = tan 46.2°
(h-6)/150 = 1.04
h-6 = 156.42
h = 162.42
Well, well, well, we have a height-hunting surveyor here! Let's embrace our inner math clown and solve this riddle!
So, the surveyor is standing 150 feet away from the tree, and the angle of elevation to the top of the tree is 46.2 degrees. We also know that the surveyor's eye level is 6 feet above the ground. Quite the precise measurements!
Now, to find the height of the tree, we need to break out our mathematical magic. We can use a little bit of trigonometry here. Specifically, the tangent function.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In our case, the opposite side is the height of the tree, and the adjacent side is the distance from the surveyor to the tree. So, we can set up the following equation:
tan(46.2) = (height of the tree) / (150 + 6)
Now, let's do a little calculation dance. We can solve for the height of the tree:
(height of the tree) = tan(46.2) * (150 + 6)
Grab your calculator and let's crunch those numbers! Once you've calculated it, you'll know the height of the tree to the nearest foot. Good luck, height-hunting surveyor!
To find the height of the tree, we can use trigonometry. We have the length of the adjacent side (150ft) and the measure of the angle of elevation (46.2 degrees).
Let's start by drawing a diagram:
* T (top of the tree)
/
/|
/ |
/ |
/ |
/θ |
/ |
/ |
S /________| G (surveyor's position)
In this diagram, G represents the surveyor's position, T represents the top of the tree, and S represents the base of the tree. The angle θ is the angle of elevation.
We can consider the right triangle GST, where GS represents the horizontal distance from the surveyor to the base of the tree, and ST represents the height of the tree.
Now, we have the adjacent side (GS) and the angle θ. We can use the trigonometric function tangent (tan) to find the height of the tree.
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is ST and the adjacent side is GS.
So, we can use the formula:
tan(θ) = ST / GS
In our case, tan(46.2 degrees) = ST / 150ft
To find ST, rearrange the formula:
ST = tan(46.2 degrees) * 150ft
Now, calculate ST:
ST ≈ tan(46.2 degrees) * 150ft
Using a scientific calculator, we find:
ST ≈ 149.9ft
However, we need to consider that the surveyor's eye level is 6ft above the ground. So, we add that height to the calculated value of ST to get the final height of the tree:
Height of the tree = ST + eye level height
Height of the tree = 149.9ft + 6ft
Height of the tree ≈ 155.9ft
Therefore, the height of the tree to the nearest foot is approximately 156 feet.
If the height is h, then
(h-6)/150 = tan 46.2°