We have a right triangle formed by the tower, the ground, and the line of sight from the man's eyes to the top of the tower. The height of the tower is the opposite side of the triangle, the distance from the man to the tower is the adjacent side, and the line of sight is the hypotenuse.
Given:
Height of the tower (opposite side) = 51.5 m
Height of the man (opposite side) = 1.5 m
Angle of elevation = 30°
We can use the tangent function in this scenario:
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
For the angle of elevation, the opposite side is the height of the tower minus the height of the man:
\[ \tan(30^\circ) = \frac{51.5 - 1.5}{\text{adjacent}} \]
Solve for the adjacent side (distance from the man to the tower):
\[ \text{adjacent} = \frac{51.5 - 1.5}{\tan(30^\circ)} \]
Calculate the value of the adjacent side (\( x \)):
\[ x = \frac{50}{\sqrt{3}} \]
This is approximately equal to 28.87 m.
So, the value of \( x \) is approximately 28.87 meters.