A and B are two balloons at the same height and 1000 metre apart. O is a ground observation position. Angle AOX is 45 degree 15' and angle BOX is 10 degree and 30'. Find the height of the balloons.

To find the height of the balloons, we can use trigonometry and the given angles. Let's break down the problem step by step:

Step 1: Draw a diagram
Draw a diagram with A and B representing the balloons, O representing the ground observation position, and X representing the point directly below the balloons on the ground.

Step 2: Calculate the distance from O to X
Since A and B are 1000 meters apart and X is directly below the balloons, the distance from O to X is also 1000 meters.

Step 3: Calculate the distance from A to X
To find the distance from A to X, we can use trigonometry. Given that angle AOX is 45 degrees 15 minutes, we can convert it to decimal degrees. 1 minute is equal to 1/60 of a degree, so 15 minutes is equivalent to 15/60 = 0.25 degrees.

Therefore, the angle AOX is 45.25 degrees. We can use the tangent function to calculate the distance from A to X.

tangent(angle AOX) = opposite/adjacent
tangent(45.25 degrees) = height of balloon A / distance from A to X

We can rearrange the equation to solve for the height of balloon A:
height of balloon A = distance from A to X * tangent(angle AOX)

Step 4: Calculate the distance from B to X
Using the same method as in Step 3, we can find the distance from B to X. Given that angle BOX is 10 degrees 30 minutes, we can convert it to decimal degrees. 30 minutes is equal to 30/60 = 0.5 degrees.

Therefore, the angle BOX is 10.5 degrees. Again, using the tangent function, we can calculate the distance from B to X.

tangent(angle BOX) = opposite/adjacent
tangent(10.5 degrees) = height of balloon B / distance from B to X

Rearranging the equation gives us:
height of balloon B = distance from B to X * tangent(angle BOX)

Step 5: Calculate the height of the balloons
Substitute the given values and calculate the height of each balloon separately using the formulas from Steps 3 and 4.

Once you have calculated the height of balloon A and balloon B, you will have your answer.