To solve the problem, we can use some trigonometry based on the angles of elevation and the distances involved.
Let:
- \( h \) = height of the tower
- \( b \) = distance from the base of the tower to Brigit = 15 m
- \( d \) = distance from the base of the tower to Rodger
- The angle of elevation from Brigit to the bird = \( 64^\circ \)
- The angle of elevation from Rodger to the bird = \( 37^\circ \)
First, we can find the height \( h \) of the tower using Brigit's position:
Using the formula for the tangent of an angle: \[ \tan(\theta) = \frac{h}{\text{base}} \] For Brigit: \[ \tan(64^\circ) = \frac{h}{15} \] Thus, \[ h = 15 \cdot \tan(64^\circ) \]
Calculating \( h \): \[ h = 15 \cdot \tan(64^\circ) \approx 15 \cdot 1.9626 \approx 29.439 \text{ m} \]
Next, we find the distance \( d \) from the base of the tower to Rodger: Using the angle of elevation from Rodger: \[ \tan(37^\circ) = \frac{h}{d} \] Thus, \[ d = \frac{h}{\tan(37^\circ)} \]
Calculating \( \tan(37^\circ) \): \(\tan(37^\circ) \approx 0.7536\)
Now substituting in the value of \( h \): \[ d = \frac{29.439}{0.7536} \approx 39.06 \text{ m} \]
Now we can find the distance between Brigit and Rodger: \[ \text{Distance between Brigit and Rodger} = d - b = 39.06 - 15 = 24.06 \text{ m} \]
Therefore, the distance between Brigit and Rodger is approximately \( \boxed{24.06} \text{ m} \).