The angle of elevation from a point on the ground to the top of a tree is 39.8 degrees. The angle of elevation from a point 51.7 ft farther back to the top of the tree is 19.7 degrees. Help find the height of the tree.
well, you know that
h/x = tan 39.8°
h/(x+51.7) = tan 19.7°
Now eliminate x and solve for h:
h/tan39.8° = h/tan19.7° - 51.7
1.2h = 2.8h - 51.7
h = 32.3
check my math, but that's the method to my madness.
To find the height of the tree, we can use the concept of trigonometry.
Let's denote the height of the tree as "h" and the distance from the first point to the tree as "x".
From the given information, we have two right-angled triangles: one at the first point and another at the second point.
In the first triangle:
1. The angle of elevation is 39.8 degrees from the ground to the top of the tree.
2. The opposite side is the height of the tree, h.
3. The adjacent side is the distance from the first point to the tree, x.
In the second triangle:
1. The angle of elevation is 19.7 degrees from a point 51.7 ft farther back to the top of the tree.
2. The opposite side is the height of the tree, h.
3. The adjacent side is the distance from the second point to the tree, x + 51.7 ft.
Using the trigonometric ratios, we can write equations based on these triangles:
For the first triangle: tan(39.8°) = h / x ------(1)
For the second triangle: tan(19.7°) = h / (x + 51.7) ------(2)
Now we can solve these equations to find the height of the tree.
First, let's solve equation (1) for h:
h = x * tan(39.8°)
Next, substitute this value back into equation (2):
tan(19.7°) = (x * tan(39.8°)) / (x + 51.7)
Now, we can solve for x by cross-multiplying:
tan(19.7°) * (x + 51.7) = x * tan(39.8°)
Expanding and rearranging the equation:
x * tan(19.7°) + 51.7 * tan(19.7°) = x * tan(39.8°)
Now, solve for x:
x * (tan(39.8°) - tan(19.7°)) = 51.7 * tan(19.7°)
x = (51.7 * tan(19.7°)) / (tan(39.8°) - tan(19.7°))
Once you calculate the value of x, substitute it back into equation (1) to find the height of the tree:
h = x * tan(39.8°)
This will give you the height of the tree.
To find the height of the tree, we can use trigonometry and create a right triangle with the given information. Let's denote the height of the tree as h, the distance from the first point to the tree as x, and the distance from the second point to the tree as x + 51.7 ft.
From the given information, we can determine the following:
1. In the first triangle:
- The angle of elevation is 39.8 degrees.
- The opposite side is the height of the tree, h.
- The adjacent side is the distance from the first point to the tree, x.
2. In the second triangle:
- The angle of elevation is 19.7 degrees.
- The opposite side is the height of the tree, h.
- The adjacent side is the distance from the second point to the tree, x + 51.7 ft.
Using the tangent function for both triangles, we can set up the following equations:
For the first triangle:
tan(39.8°) = h / x
For the second triangle:
tan(19.7°) = h / (x + 51.7)
Now, we can solve these equations to find the height of the tree, h.
1. Rearrange the first equation to solve for h:
h = x * tan(39.8°) (Equation 1)
2. Rearrange the second equation to solve for h:
h = (x + 51.7) * tan(19.7°) (Equation 2)
3. Set Equation 1 equal to Equation 2:
x * tan(39.8°) = (x + 51.7) * tan(19.7°)
4. Solve for x by isolating it:
x * tan(39.8°) - x * tan(19.7°) = 51.7 * tan(19.7°)
5. Factor out x from the left side:
x * (tan(39.8°) - tan(19.7°)) = 51.7 * tan(19.7°)
6. Solve for x:
x = (51.7 * tan(19.7°)) / (tan(39.8°) - tan(19.7°))
Now that we have the value of x, we can substitute it back into Equation 1 or Equation 2 to find the height of the tree, h:
h = x * tan(39.8°) (Equation 1)
h = (x + 51.7) * tan(19.7°) (Equation 2)
Calculating the values using these equations will give us the height of the tree.