Two tracking stations 50 miles apart sight a weather ballon. The angle of the ballon from one of the stations is 45° and from the other is 60°. What is height of the balloon?

If the balloon is between the stations

h/x = tan45°
h/(50-x) = tan60°

h/tan45° = 50 - h/tan60°
h = 50 - h/√3
h = 50/(1+1/√3) = 31.7

There is another answer if both stations are on the same side of the balloon.

Well, it seems like the weather balloon is making quite the angle spectacle! Let's solve this height riddle.

We can start by drawing a triangle with the two tracking stations and the weather balloon as its vertices. Let's call the height of the balloon "h."

Now, since the distance between the tracking stations is 50 miles, let's label that as "d." With that in mind, we can denote the angle from one tracking station as 45° and the other as 60°.

Alright, time to clown around with some math! We can use trigonometry to find the height.

For the 45° angle, we'll use the tangent function: tan(45°) = h/d.

For the 60° angle, we'll use the tangent function as well: tan(60°) = h/(d + 50).

Now, we can plug in some values and have some math fun!

tan(45°) = h/50
tan(60°) = h/(d + 50)

Double-checking our trusty trigonometric tables, we find that tan(45°) equals 1.

To make this equation great again, we'll solve it for h:
h = 50

For the next equation, tan(60°) equals the square root of 3, or approximately 1.732.

So, h/(d + 50) = 1.732.

Now, we can substitute the values.
1.732 = h/(50 + 50)
1.732 = h/100

Rearranging the equation to solve for h:

h = 1.732 * 100

Alright, time for the grand finale!

h = 173.2 feet

So, the height of the weather balloon is approximately 173.2 feet. Now that's some high-flying humor!

To find the height of the balloon, we can use trigonometry and create two right triangles.

Let:
- A be one tracking station
- B be the other tracking station
- C be the balloon
- AB be the distance between the two tracking stations, which is 50 miles
- AC be the height of the balloon
- ∠CAB be the angle of the balloon from the tracking station A, which is 45°
- ∠CBA be the angle of the balloon from the tracking station B, which is 60°

We can use the tangent function to find the height of the balloon in relation to the distance between the two tracking stations.

In triangle ABC, we can use the tangent of ∠CAB to find AC:
tan(∠CAB) = AC / AB

tan(45°) = AC / 50 miles

AC = tan(45°) * 50 miles

AC ≈ 1 * 50 miles

AC ≈ 50 miles

Now, we can find the length of BC using triangle BAC:

In triangle BAC, we can use the tangent of ∠CBA to find BC:
tan(∠CBA) = BC / AB

tan(60°) = BC / 50 miles

BC ≈ √3 * 50 miles

BC ≈ 1.732 * 50 miles

BC ≈ 86.6 miles

To find the height of the balloon, we need to find the difference between AC and BC:

Height of the balloon = AC - BC

Height of the balloon ≈ 50 miles - 86.6 miles

Height of the balloon ≈ -36.6 miles

Since the height cannot be negative, it means there's an error in the calculation or the given angles. Please recheck the given angles and try again.

To find the height of the balloon, we can create a triangle with the tracking stations and the balloon. Let's call the angles at the tracking stations A and B, and the distance between the tracking stations d.

Using basic trigonometry, we can use the tangent function to relate the angles and sides of the triangle.
tan(A) = height / distance from station A
tan(B) = height / distance from station B

In this case, tan(45°) = height / distance from station A
tan(60°) = height / distance from station B

Given that the distance between the tracking stations is 50 miles, we can calculate the distances from each station to the balloon using the trigonometric ratios:

distance from station A = 50 * tan(45°)
distance from station B = 50 * tan(60°)

Now, let's substitute these values back into our equations:

height = distance from station A * tan(A)
height = (50 * tan(45°)) * tan(45°)

To calculate the value of height, we'll first find the value of tan(45°) using a scientific calculator or refer to a table of trigonometric values. tan(45°) is equal to 1.

Now we can substitute this value back into our equation:

height = (50 * 1) * tan(45°)

Simplifying further, we get:

height = 50 * tan(45°)
height = 50 * 1
height = 50

Therefore, the height of the balloon is 50 miles.