From a police helicopter flying at 2,400 ft, a stolen car is sighted at an angle of depression of 73°. Find the distance of the car (to the nearest foot) from a point directly below the helicopter.
734

We can solve this problem by using trigonometry, specifically the tangent function. Here's a step-by-step explanation:

### Problem Overview:
- Height of the helicopter: \( h = 2400 \) ft
- Angle of depression: \( \theta = 73^\circ \)
- We need to find the horizontal distance \( d \) from the car to a point directly below the helicopter.

### Applying the Tangent Function:

The tangent of an angle in a right triangle is given by:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

In this case:
- Opposite side = height of the helicopter = 2400 ft
- Adjacent side = horizontal distance \( d \) (what we are solving for)

So, the equation becomes:

\[
\tan(73^\circ) = \frac{2400}{d}
\]

Now, solving for \( d \):

\[
d = \frac{2400}{\tan(73^\circ)}
\]

Using a calculator to find \( \tan(73^\circ) \):

\[
\tan(73^\circ) \approx 3.2709
\]

Now substitute this value into the equation:

\[
d = \frac{2400}{3.2709} \approx 733.66 \, \text{ft}
\]

Rounding to the nearest foot:

\[
d \approx 734 \, \text{ft}
\]

### Final Answer:
The car is approximately **734 feet** away from the point directly below the helicopter.

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Would you like further details or have any questions about this solution?

Here are 5 related questions to deepen your understanding:
1. How would the horizontal distance change if the angle of depression were smaller?
2. How do we handle situations where the angle of elevation is given instead of depression?
3. Can you solve the problem using sine or cosine functions instead of tangent?
4. What is the relationship between the angle of depression and angle of elevation in these problems?
5. How would the result differ if the height of the helicopter increased?

**Tip:** Always check if you're dealing with the angle of depression or elevation, as this changes the way you approach the problem.

From a police helicopter flying at 2,400 ft, a stolen car is sighted at an angle of depression of 73°. Find the distance of the car (to the nearest foot) from a point directly below the helicopter.

Learn how to find the distance of a car from a point directly below a helicopter using trigonometry. In this example, a helicopter is flying at 2,400 feet, and the car is sighted at an angle of depression of 73°. This step-by-step solution explains how to apply the tangent function in right triangles to solve the problem, suitable for Grades 9-11.

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