To solve these problems, we can use basic trigonometry principles, specifically the tangent function.
1. Let's first consider the problem about the ranger's tower and the tall tree.
From the top of the tower, the ranger looks up to the top of the tree. This forms an angle of elevation of 28 degrees.
Drawing a diagram, you can visualize a right-angled triangle with the tower, the top of the tree, and the ranger at the top vertex.
The side opposite the 28-degree angle is the height of the tree (h), and the adjacent side is the distance between the tower and the tree (44m).
Using the tangent function, we have the equation:
tan(28°) = h/44
To find h, multiply both sides of the equation by 44:
h = 44 * tan(28°)
Using a calculator, we can find:
h ≈ 21.7 meters
Therefore, the tree is approximately 21.7 meters tall.
2. Now let's solve the problem with the engineer and the tower.
The engineer measures the angle of elevation to the top of the tower from two different points, both due east of the town.
We have two right-angled triangles, one from each location.
In the first triangle, the engineer is directly east of the tower, and the angle of elevation is 52 degrees.
In the second triangle, the engineer is 47 meters east of the tower, and the angle of elevation is 31 degrees.
Let's denote the height of the tower as h and the distance from the first point to the tower as x. Therefore, the distance from the second point to the tower is (x + 47)m.
Using the tangent function, we can set up two separate equations:
tan(52°) = h/x
tan(31°) = h/(x + 47)
We can now solve this system of equations to find the value of h.
Dividing the second equation by the first equation, we eliminate h:
tan(31°) / tan(52°) = (h/(x + 47)) / (h/x)
tan(31°) / tan(52°) = (x / (x + 47))
Now, we can solve for x. Rearranging the equation:
x / (x + 47) = tan(31°) / tan(52°)
Cross-multiplying:
x * tan(52°) = (x + 47) * tan(31°)
Expanding:
x * 0.6658 = (x + 47) * 0.6009
Simplifying the equation:
0.6658x = 0.6009x + 28.2483
0.065x = 28.2483
x ≈ 434.6
Substituting this value back into either of the original equations (e.g., x/tan(52°) = h), we can find the height of the tower.
Using a calculator, we can solve for h:
h ≈ 434.6 / tan(52°)
h ≈ 434.6 / 1.2799
h ≈ 339.8 meters
Thus, the height of the tower is approximately 339.8 meters.