A drone is hovering 800.5 m above a straight road. The drone operator observes that the angle of depression of a car to the north and a bus to the south of the drone's position are 25°12' and 40°45', respectively. How far is the car from the bus? Draw the triangle

To solve this problem, we can use trigonometry. The key is to form two right-angled triangles with the drone's position as the common vertex and the car and bus as the opposite vertices along the road. We'll then determine the distances from the drone to the car and the bus, and sum these to find the distance between the car and the bus.

### Step-by-step solution:

1. **Convert the angles to decimal form:**
   - For the car, the angle of depression is \( 25^\circ 12' \). Convert to decimal:
     \[
     25^\circ + \frac{12'}{60} = 25.2^\circ
     \]
   - For the bus, the angle of depression is \( 40^\circ 45' \). Convert to decimal:
     \[
     40^\circ + \frac{45'}{60} = 40.75^\circ
     \]

2. **Understanding the geometry:**
   - The drone is 800.5 m above the road, forming a vertical line.
   - The car and the bus are along the road at two different distances from the point directly under the drone.
   - The angle of depression (from the drone's position) forms a right-angled triangle with the road.

3. **Apply trigonometry (tangent function):**
   - In a right triangle, the tangent of the angle is the ratio of the opposite side (the height of the drone, 800.5 m) to the adjacent side (the distance along the road, which we want to find).

   - For the car:
     \[
     \tan(25.2^\circ) = \frac{800.5}{d_1} \quad \text{where } d_1 \text{ is the distance from the drone to the car.}
     \]
     Solving for \( d_1 \):
     \[
     d_1 = \frac{800.5}{\tan(25.2^\circ)}
     \]
     \[
     d_1 \approx \frac{800.5}{0.471} \approx 1699.36 \, \text{m}
     \]

   - For the bus:
     \[
     \tan(40.75^\circ) = \frac{800.5}{d_2} \quad \text{where } d_2 \text{ is the distance from the drone to the bus.}
     \]
     Solving for \( d_2 \):
     \[
     d_2 = \frac{800.5}{\tan(40.75^\circ)}
     \]
     \[
     d_2 \approx \frac{800.5}{0.861} \approx 929.46 \, \text{m}
     \]

4. **Find the total distance:**
   - The car is to the north and the bus to the south, so the total distance between the car and the bus is:
     \[
     \text{Total distance} = d_1 + d_2 = 1699.36 + 929.46 \approx 2628.82 \, \text{m}
     \]

Thus, the distance between the car and the bus is approximately **2628.82 meters**.

### Drawing the Triangle:
The drone is at the top of a vertical line 800.5 m above the road. Two right-angled triangles extend from the drone's vertical position down to the road, one for the car and one for the bus. The angles at the drone's position are the angles of depression (25.2° for the car and 40.75° for the bus). The adjacent sides (along the road) are the distances we calculated for the car and the bus.

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Here are 5 related questions to expand on the concept:
1. How would the solution change if the height of the drone were different?
2. What would happen if the drone’s height was doubled?
3. Can you find the distance between the drone and both the car and bus?
4. What if the road were inclined instead of flat? How would the angles change?
5. How would this problem look if the drone were positioned between the car and bus?

**Tip:** Always remember to convert angles from degrees and minutes to decimal form when using trigonometric functions.

A drone is hovering 800.5 m above a straight road. The drone operator observes that the angle of depression of a car to the north and a bus to the south of the drone's position are 25°12' and 40°45', respectively. How far is the car from the bus? Draw the triangle.

Solve a trigonometry problem involving a drone hovering 800.5 m above a road. The angles of depression to a car and a bus are 25°12' and 40°45', respectively. Learn how to calculate the distance between the car and bus using the tangent function. Suitable for high school students in Grades 9-12.

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