To find the angle of elevation to the top of the building from Naledi's position, we can use the tangent function from trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side (height of the building) to the adjacent side (distance from the base of the building).
Let:
- \( h \) = height of the building
- \( d \) = distance from the base of the building = 42 m
- \( d_t \) = distance to the top of the building = 63 m
- \( \theta \) = angle of elevation
Using the Pythagorean theorem, we can express the relationship between these values:
\[ d_t^2 = h^2 + d^2 \]
Substituting the known values:
\[ 63^2 = h^2 + 42^2 \]
Calculating the squares:
\[ 3969 = h^2 + 1764 \]
Now, we can isolate \( h^2 \):
\[ h^2 = 3969 - 1764 = 2205 \]
Next, take the square root to find \( h \):
\[ h = \sqrt{2205} \approx 47.0 \text{ m (rounded to one decimal place)} \]
Now, we can find the angle of elevation \( \theta \) using the tangent function:
\[ \tan(\theta) = \frac{h}{d} = \frac{47}{42} \]
Calculating \( \tan(\theta) \):
\[ \tan(\theta) \approx 1.119 \]
Now, to find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(1.119) \approx 48.37^\circ \]
Thus, the angle of elevation to the top of the building is approximately \( 48.37^\circ \).