To find the distance between the zoo and the library, we can use trigonometric relationships based on the angles given and the height of the helicopter.
The height of the helicopter above the ground is given as:
\[ h = 300 \sqrt{3} \text{ feet} \]
- Finding the horizontal distance to the zoo:
The angle between the line from the helicopter to the zoo and the ground is 60 degrees. We can use the tangent function to find the horizontal distance (denote it as \(d_z\)) from the point directly underneath the helicopter to the zoo:
\[ \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d_z} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{300 \sqrt{3}}{d_z} \]
From this, we can solve for \(d_z\):
\[ d_z = \frac{300 \sqrt{3}}{\sqrt{3}} = 300 \text{ feet} \]
- Finding the horizontal distance to the library:
The angle between the line from the helicopter to the library and the ground is 30 degrees. We can find the horizontal distance (denote it as \(d_l\)) from the point directly underneath the helicopter to the library:
\[ \tan(30^\circ) = \frac{h}{d_l} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{300 \sqrt{3}}{d_l} \]
From this, we can solve for \(d_l\):
\[ d_l = 300 \sqrt{3} \cdot \sqrt{3} = 300 \cdot 3 = 900 \text{ feet} \]
- Calculating the distance between the zoo and the library:
Now, the total distance between the zoo and the library can be found by subtracting the horizontal distance to the zoo from the horizontal distance to the library:
\[ \text{Distance} = d_l - d_z = 900 \text{ feet} - 300 \text{ feet} = 600 \text{ feet} \]
Thus, the distance between the zoo and the library is:
\[ \boxed{600} \text{ feet} \]
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