A weather balloon is directly west of two observing stations that are 10 miles apart. The angles of elevation of the balloon from the two stations are 17.6 degrees and 78.2 degrees. How high is the balloon?

To find the height of the balloon, we can use the trigonometric concept of tangent.

Let's assume the height of the balloon is "h" (in miles).

From the first observing station, the angle of elevation is 17.6 degrees. This means that the tangent of the angle is equal to opposite (h) divided by adjacent (10 miles).

Therefore, tan(17.6) = h / 10

Rearranging the equation, we have:
h = 10 * tan(17.6)

Using a calculator, we find that the height of the balloon from the first observing station is approximately 3.10 miles.

Similarly, from the second observing station, the angle of elevation is 78.2 degrees. Using the same concept, we have:
h = 10 * tan(78.2)

Using a calculator, we find that the height of the balloon from the second observing station is approximately 31.47 miles.

Since we can assume that both angles of elevation are measuring the same height, we can take the average of the two heights:
Average height = (3.10 + 31.47) / 2

Calculating the average height, we find that the height of the balloon is approximately 17.28 miles.

Therefore, the height of the balloon is approximately 17.28 miles.

To find the height of the balloon, we can use trigonometry and the concept of similar triangles. Let's call the distance from the first station to the balloon 'x' and the distance from the second station to the balloon 'y'.

From the first station, we have an angle of elevation of 17.6 degrees. This means that the tangent of the angle is equal to the height of the balloon divided by the distance 'x'. So, we have:

tan(17.6) = height of balloon / x

Similarly, from the second station with an angle of elevation of 78.2 degrees:

tan(78.2) = height of balloon / y

From the problem, we know that the two stations are 10 miles apart, so we have:

x + y = 10

Now, we can solve this system of equations to find the height of the balloon.

First, rearrange the equation x + y = 10 to express one variable in terms of the other. Let's solve for x:

x = 10 - y

Now substitute this value of x in the first equation:

tan(17.6) = height of balloon / (10 - y)

Cross multiply to get:

height of balloon = tan(17.6) x (10 - y)

Now substitute this value in the second equation:

tan(78.2) = (height of balloon) / y

Cross multiply to get:

height of balloon = tan(78.2) x y

Since we have two expressions for the height of the balloon, we can equate them:

tan(17.6) x (10 - y) = tan(78.2) x y

Now we can solve this equation to find the value of y, which represents the distance from the second station to the balloon.

Tan78.2=h/x. h=x(tan78.2)

Tan17.6=h/(x+10). h=(x+10)(tan17.6)

Need to solve for x and then go back for your height.