From a police helicopter flying at 2,900 ft, a stolen car is sighted at an angle of depression of 76°. Find the distance of the car (to the nearest foot) from a point directly below the helicopter.
724

To solve this problem, we can use trigonometry, specifically the tangent function, which relates the angle of depression, the height of the helicopter, and the horizontal distance from the point directly below the helicopter to the stolen car.

### Step-by-step solution:
1. Let:
   - \( h = 2900 \, \text{ft} \) be the height of the helicopter.
   - \( \theta = 76^\circ \) be the angle of depression.
   - \( d \) be the horizontal distance from the car to the point directly below the helicopter.

2. The tangent function is defined as:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
   \]
   Therefore:
   \[
   \tan(76^\circ) = \frac{2900}{d}
   \]

3. Solving for \( d \):
   \[
   d = \frac{2900}{\tan(76^\circ)}
   \]

4. Using a calculator to find \( \tan(76^\circ) \):
   \[
   \tan(76^\circ) \approx 4.0108
   \]

5. Now, substitute the value of \( \tan(76^\circ) \) into the equation:
   \[
   d = \frac{2900}{4.0108} \approx 723.42
   \]

6. Round the result to the nearest foot:
   \[
   d \approx 723 \, \text{ft}
   \]

So, the distance of the car from the point directly below the helicopter is approximately **723 feet**.

Would you like more details or have any questions about the steps?

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Here are 5 related questions:
1. What is the horizontal distance if the angle of depression was 60° instead of 76°?
2. How would the height of the helicopter affect the horizontal distance if it was flying at 5,000 ft?
3. What is the relationship between the angle of depression and the horizontal distance?
4. Can the same method be used to calculate the distance if the object was above the helicopter, like in a drone scenario?
5. How would you calculate the actual straight-line distance between the helicopter and the car?

**Tip:** The tangent function is useful in right-angled triangle problems, especially when dealing with heights and horizontal distances in real-life applications like this.

From a police helicopter flying at 2,900 ft, a stolen car is sighted at an angle of depression of 76°. Find the distance of the car (to the nearest foot) from a point directly below the helicopter.

Learn how to use trigonometry to calculate the horizontal distance of a stolen car from a police helicopter flying at 2,900 feet, given an angle of depression of 76°. This step-by-step solution explains the use of the tangent function and is suitable for students in Grades 9-11.

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