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?
To find the height of the balloon, we can use trigonometry and the concept of similar triangles.
Let's label the two observers as A and B, and the height of the hot-air balloon as h.
From observer A's perspective, the angle of elevation to the balloon is 33 degrees. Similarly, from observer B's perspective, the angle of elevation to the balloon is 37 degrees.
First, let's draw a diagram to visualize the situation:
```
A
\
\
\
o Balloon
/
/
/
B
```
We want to find the height of the balloon (h). Since the two observers are facing each other, the line AB represents the ground, and the line segment connecting the balloon to the ground represents the height (h) of the balloon.
Now, we can use the concept of similar triangles to find the height of the balloon.
In triangle AOB, ∠AOB is a right angle since AB represents the level ground.
In triangle AOC, ∠AOC is the angle of elevation (33 degrees) from observer A.
In triangle BOD, ∠BOD is the angle of elevation (37 degrees) from observer B.
Since the alternate interior angles in a transversal are congruent, ∠AOC and ∠BOD are congruent.
Using the Law of Sines, we can set up the following proportion:
sin(∠AOC) / OC = sin(∠BOD) / OD
sin(33) / OC = sin(37) / OD
To find OC and OD, we need to calculate the distances AO and BO in terms of OC and OD.
AO = OC * tan(∠AOC)
BO = OD * tan(∠BOD)
Now we can substitute these values into the proportion:
sin(33) / OC = sin(37) / OD
sin(33) / (OC * tan(∠AOC)) = sin(37) / (OD * tan(∠BOD))
Rearranging the equation, we get:
h / OC = h / OD
Cross-multiplying:
h * OD = h * OC
Dividing the equation by h:
OD = OC
Since AO + BO = AB (the distance between the observers), we can substitute OC + OD for AB in the equation:
OC * tan(∠AOC) + OD * tan(∠BOD) = AB
OC * tan(∠AOC) + OC * tan(∠BOD) = AB
OC * (tan(∠AOC) + tan(∠BOD)) = AB
Now, we can substitute the known values into the equation:
OC * (tan(33) + tan(37)) = 1.75 miles
Solving for OC:
OC = 1.75 miles / (tan(33) + tan(37))
Once we have OC, we can find the height of the balloon by using the equation:
h = OC * tan(∠AOC)
Substituting the known values:
h = OC * tan(33)
Using the values obtained from the calculations, the height of the balloon can be determined.
To determine the height of the hot-air balloon above the ground, we can use the concept of similar triangles. Let's denote the height of the balloon as 'h'.
Step 1: Draw a diagram to represent the situation. Label the two observers as A and B, the distance between them as 1.75 miles, and the height of the balloon as 'h'. Also, mark the angles of elevation as 33° and 37°.
Step 2: Recognize that we have two right triangles, ACD and BCD, with corresponding angles of 33° and 37°. The side opposite the angle of elevation represents the height of the balloon ('h').
Step 3: Apply the concept of tangent to find the height of the balloon in relation to the distance between the observers.
In triangle ACD:
tan(33°) = h/1.75
In triangle BCD:
tan(37°) = h/1.75
Step 4: Rearrange the equations to solve for 'h'.
h = 1.75 * tan(33°)
h = 1.75 * tan(37°)
Step 5: Use a calculator to find the values of tan(33°) and tan(37°).
tan(33°) ≈ 0.6494
tan(37°) ≈ 0.7536
Step 6: Substitute the values into the equations to find the height of the balloon.
h ≈ 1.75 * 0.6494 ≈ 1.137 meter (approx.)
h ≈ 1.75 * 0.7536 ≈ 1.319 meter (approx.)
Therefore, the hot-air balloon is approximately 1.137-1.319 meters above the ground.