weather balloon is directly west of two observing stations that are 10 mi 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?

Make the following diagram

Draw a base line , label the balloon as B and above the base line
Label the point directly below the balloon A on the base line
label the two stations as C and D so that
angle ACB = 78.2 and angle ADB = 17.6
Now in triangle BCD , CD = 10 ,
angle BCD = 101.8 , (exterior angle theorem)
and angle CBD = 60.6° (angle sum of triangle theorem)

Using the sine law BC/sin17.6 = 10/sin60.6
you can find BC, which is then the hypotenuse of
triangle ABC
then use
sin 78.2 = height/BC to find the height

To find the height of the weather balloon, we can use trigonometry and the concept of similar triangles.

Let's assume the height of the balloon is h.

From the right triangle formed by the first observing station, we can write the following equation using tangent:

tan(17.6 degrees) = h / (10 miles)

Rearranging the equation, we get:

h = tan(17.6 degrees) * 10 miles

Using a calculator, we can find the value of tan(17.6 degrees) ≈ 0.3146.

h ≈ 0.3146 * 10 miles
h ≈ 3.146 miles

Therefore, the height of the weather balloon is approximately 3.146 miles.

To find the height of the weather balloon, we can use trigonometry and assume the distance between the weather balloon and the two observing stations is the same.

Let's call the height of the balloon 'h'.

From the first observation station, the angle of elevation to the balloon is 17.6 degrees. This means that if we draw a right triangle with the balloon at the top, the observer on the ground, and a line connecting them, the angle between the ground and the line connecting the observer to the balloon is 17.6 degrees.

Similarly, from the second observation station, the angle of elevation to the balloon is 78.2 degrees. Again, if we draw a right triangle with the balloon at the top, the observer on the ground, and a line connecting them, the angle between the ground and the line connecting the observer to the balloon is 78.2 degrees.

Now, we can use the tangent function to find the height of the balloon.

From the first observation station, we have the equation:

tan(17.6 degrees) = h / x, where x is the distance between the stations (10 miles).

Solving for h, we get:

h = x * tan(17.6 degrees)

Similarly, from the second observation station, we have the equation:

tan(78.2 degrees) = h / x

Solving for h again, we get:

h = x * tan(78.2 degrees)

Since we assumed that the distance between the balloon and the two observing stations is the same (x is the same), we can set the two equations equal to each other:

x * tan(17.6 degrees) = x * tan(78.2 degrees)

Now, we can solve for x. Divide both sides by x:

tan(17.6 degrees) = tan(78.2 degrees)

Using a scientific calculator, we find:

0.315586 = 6.01482

Since the two sides are not equal, we made an error somewhere in our calculations or assumptions. Please double-check the information and calculations provided to proceed.