two observers 3 miles apart and facing each other find that the angles of the elevation of a balloon in the same vertical plane with themselves are 28degree and 31 degree respectively.find the distance from the balloon to the observer located at the 31 angle?

Label the observers A and B, with point C on the line AB, directly below the balloon, at point D.

If x is the distance CB, then since AC+CB=3,

(3-x)tan28° = xtan31°
x = 1.4 miles.

Now, you want the distance DB, which is

x/DB = cos31°

DB = x/cos31° = 1.4/.857 = 1.63 miles

To find the distance from the balloon to the observer located at the 31-degree angle, we can use trigonometry and the concept of similar triangles.

Let's assume the distance from the balloon to the observer at the 28-degree angle is x.

Using the concept of similar triangles, we can set up the following equation:

tan(28 degrees) = height of the balloon / x

We can rearrange this equation to solve for the height of the balloon:

height of the balloon = x * tan(28 degrees)

Now, let's consider the observer at the 31-degree angle. The distance between the two observers is 3 miles, and the distance from the balloon to the observer at the 31-degree angle can be represented as (3 - x).

Using the concept of similar triangles again, we can set up the following equation:

tan(31 degrees) = height of the balloon / (3 - x)

Rearranging this equation, we can solve for the height of the balloon in terms of (3 - x):

height of the balloon = (3 - x) * tan(31 degrees)

Since the height of the balloon should be the same, we can equate the two expressions for the height of the balloon:

x * tan(28 degrees) = (3 - x) * tan(31 degrees)

Now, we can solve this equation to find the value of x:

x * tan(28 degrees) = (3 - x) * tan(31 degrees)

x * 0.532 = (3 - x) * 0.600

0.532x = 1.800 - 0.600x

0.532x + 0.600x = 1.800

1.132x = 1.800

x = 1.591

So, the distance from the balloon to the observer located at the 31-degree angle is approximately (3 - 1.591) miles, which is about 1.409 miles.

To solve this problem, we can use the concept of trigonometry and the properties of triangles. Here's how you can find the distance from the balloon to the observer located at the 31-degree angle:

1. Draw a diagram: Start by drawing a diagram to visualize the given information. Draw two observers facing each other, 3 miles apart, and label them as A and B. Now draw a balloon above them in the same vertical plane, labeling it as C. Connect observer A to the balloon C and observer B to the balloon C with lines.

2. Use the tangent function: Since we have angles of elevation, we can use the tangent function to relate the angles to the distances. Remember that the tangent of an angle is defined as the opposite side divided by the adjacent side.

tan(angle) = opposite/adjacent

3. Calculate the distances: Let's assume the distance from observer A to the balloon is x miles. Therefore, the distance from observer B to the balloon would be (3 - x) miles (since they are 3 miles apart).

Now, we can set up the tangent equations:

tan(28°) = x/AC (eq. 1)
tan(31°) = (3 - x)/BC (eq. 2)

4. Solve the equations: We need to solve equations 1 and 2 simultaneously to find the values of x and (3 - x).

Rearrange equation 1 to solve for AC:
AC = x/tan(28°)

Substitute this value into equation 2:
tan(31°) = (3 - x) / (x / tan(28°))
Simplify:
tan(31°) = (3 - x) * (tan(28°) / x)

Now, solve this equation for x:
x * tan(31°) = (3 - x) * tan(28°)
x * tan(31°) = 3 * tan(28°) - x * tan(28°)
x * (tan(31°) + tan(28°)) = 3 * tan(28°)
x = (3 * tan(28°)) / (tan(31°) + tan(28°))

Calculate this value using a calculator or by substituting the angle values in degrees from a trigonometric table.

5. Find the distance to the balloon: Now that you have the value of x, you can calculate the distance from observer B to the balloon using the equation (3 - x).

Distance from observer B = 3 - x

Substitute the calculated value of x into this equation to find the distance from observer B to the balloon.

That's it! You have now found the distance from the balloon to the observer located at the 31-degree angle.