To find the height of the building, we can use trigonometry. The situation described can be modeled using a right triangle.
Let's denote:
- \( h \) as the height of the building,
- \( s \) as the length of the shadow (150 meters),
- \( \theta \) as the angle between the line joining the top of the building to the end of the shadow and the shadow (45 degrees).
In this right triangle, we can use the tangent function, which relates the opposite side (the height of the building) to the adjacent side (the length of the shadow):
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{s} \]
Given that \( \theta = 45^\circ \) and \( s = 150 \) meters, we have:
\[ \tan(45^\circ) = 1 \]
Thus, substituting into the equation:
\[ 1 = \frac{h}{150} \]
From this equation, we can solve for \( h \):
\[ h = 150 \cdot 1 = 150 \text{ meters} \]
The height of the building is therefore 150 meters.
Thus, the answer is 150 m.