a hot-air balloon is seen in the sky simultaneously by two observers standing a t two different points on the ground. (the 2 people are facing each other and the balloon is in between them in the air.)they are 1.75 miles apart on level ground. The angles of elevation are 33 and 37 respectively. how high above the ground is the balloon?

Question

a hot-air balloon is seen in the sky simultaneously by two observers standing a t two different points on the ground. (the 2 people are facing each other and the balloon is in between them in the air.)they are 1.75 miles apart on level ground. The angles of elevation are 33 and 37 respectively. how high above the ground is the balloon?

Answer 1

make a diagram

let the base be BC, where angleB= 37 and angle C = 33°
From the top vertex A drop an altitude to meet BC at D
let AD = h, BD = x, then DC = 1.75-x

you now have two right-angled triangles
from ABD,
tan 37 = h/x
h = xtan37

from ADC
tan 33 = h/(1.75-x)
h = tan33(1.75-x)

then x tan37 = 1.75tan33 - xtan33
xtan37 - xtan33 = 1.75tan33
x(tan37-tan33) = 1.75tan33
x = 1.75tan33/(tan37-tan33)

once you have that evaluated, put that x value into
h = xtan37

you do all the button-pushing.

Answer 2

i got up to the very last step but when i plugged my x value into h=xtan37 i just got decimals, like .97131.. what is the answer supposed to be?

Answer 3

To determine the height of the hot-air balloon above the ground, we can use trigonometry. Let's assume the height of the balloon is denoted by 'h'.

Step 1: Draw a diagram to visualize the situation.

B (Balloon)
|\
| \
| \
| \
| \
| \
| \
| \
| \
| \
A1___|__________\___A2 (Observers)
D 1.75 miles

Step 2: Label the given information.
- The distance between the two observers is 1.75 miles.
- The angle of elevation from observer A1 to the balloon is 33 degrees.
- The angle of elevation from observer A2 to the balloon is 37 degrees.

Step 3: Identify the right triangles within the diagram.
- Triangle A1BD is a right triangle, where D denotes the position of the balloon.
- Triangle A2BD is also a right triangle.

Step 4: Write down the trigonometric ratios for each triangle.
For triangle A1BD:
Tan(33) = h / x
Where 'x' denotes the horizontal distance from observer A1 to the balloon.

For triangle A2BD:
Tan(37) = h / (1.75 - x)
Where (1.75 - x) denotes the horizontal distance from observer A2 to the balloon.

Step 5: Solve the equations simultaneously to find 'h'.
Using the given values for both angles of elevation (33 and 37 degrees), we can solve the above equations simultaneously.

Tan(33) = h / x → Equation 1
Tan(37) = h / (1.75 - x) → Equation 2

We can rewrite Equation 1 as:
x = h / Tan(33)

We can then substitute this value of 'x' into Equation 2:
Tan(37) = h / (1.75 - (h / Tan(33)))

Simplifying Equation 2 further, we get:
Tan(37) = h / (1.75 - h / Tan(33))

Step 6: Solve the equation to find 'h'.
Using any calculator or online tool, plug in the values of Tan(37) and Tan(33) to find 'h'.

Once you have calculated the value of 'h', that will be the height of the hot-air balloon above the ground.

Answer 4

To find the height of the balloon above the ground, we can use trigonometry, specifically tangent function.

Let's denote the height of the balloon as h.

First, let's consider one observer at point A. The angle of elevation from point A to the balloon is 33 degrees. We can calculate the distance from point A to the balloon by using the tangent function:

tan(33) = h / x
where x is the distance between point A and the balloon.

Similarly, for the observer at point B, the angle of elevation is 37 degrees. The distance from point B to the balloon can be calculated using the tangent function as well:

tan(37) = h / (1.75 - x)
where (1.75 - x) is the distance between point B and the balloon, considering they are 1.75 miles apart on level ground.

Now we have two equations:

tan(33) = h / x -----------(1)
tan(37) = h / (1.75 - x) -----------(2)

We can solve these two equations simultaneously to find the value of h.

Rearrange equation (1) to solve for x:

x = h / tan(33) -----------(3)

Substitute equation (3) into equation (2):

tan(37) = h / (1.75 - (h / tan(33)))

Simplify the equation:

tan(37) = h / (1.75 - h / tan(33))

Now, you can solve this equation to find the value of h.

Plug in the values for tangent of 33 degrees and 37 degrees into the equation and solve for h:

tan(37) = h / (1.75 - h / tan(33))

Using a scientific calculator to calculate the tangent values and solve for h will give you the height of the balloon above the ground.