A hot-air balloon is sighted at the same time by 2 friends who are
1.0 mile apart on the same side of the balloon. The angles of
elevation of the balloon from the 2 friends are 20.5° and 25.5°.
How high is the balloon?
AAAaannndd the bot gets it wrong yet again!
h cot20.5° - h cot25.5° = 1
h = 1.73 mi
painful
This is totally WRONG by the bot.
method1:
Make your sketch and calculate all the angles
by the sine-law:
1/sin5 = d/sin20.5
d = sin20.5/sin5 <----- the hypotenuse of the right-angled triangle
sin25.5 = h/d , where h is the height of the balloon
h = dsin25.5= (sin20.5/sin5)(sin25.5) = appr 1.73 miles high
method2:
let the height of the balloon be h miles, x be the distance on the ground
between the closer point and the base of the balloon
tan25.5 = h/x , tan20.5 = h/(1+x)
from the first: h = xtan25.5
from the 2nd: h = (1+x)tan20.5
xtan25.5 = tan20.5 + xtan20.5
x(tan25.5 - tan20.5) = tan20.5
x = tan20.5/(tan25.5 - tan20.5)
since h = xtan25.5
h = tan25.5(tan20.5/(tan25.5 - tan20.5) ) = appr 1.73 miles, just as before
To find the height of the balloon, we can use trigonometry and create a triangle with the given information.
First, let’s label the diagram. Let A be the first friend, B be the second friend, and C be the balloon. The height of the balloon will be represented by h.
We know that AC and BC represent the distances between each friend and the balloon. The angles of elevation, 20.5° and 25.5°, are the angles formed between the line connecting the friend to the balloon and the horizontal line.
Now, we can use the tangent function to find the height of the balloon:
1. tan(20.5°) = h/AC
Since the tangent of an angle is equal to the opposite side divided by the adjacent side in a right triangle, we can write the equation as:
2. tan(25.5°) = h/BC
Solving equations 1 and 2 simultaneously will give us the height of the balloon.
Let's substitute the known values into the equations:
1. tan(20.5°) = h/AC
2. tan(25.5°) = h/BC
To solve for h, we need to solve these two equations simultaneously. Here's how:
1. Rearrange equation 1 to isolate h: h = tan(20.5°) * AC
2. Substitute this value of h into equation 2: tan(25.5°) = (tan(20.5°) * AC) / BC
3. Rearrange equation 2 to solve for AC: AC = (BC * tan(25.5°)) / tan(20.5°)
4. Substitute the given distance between the friends, 1.0 mile, for BC and solve for AC: AC = (1.0 mile * tan(25.5°)) / tan(20.5°)
5. Calculate AC using a calculator. AC ≈ 1.099 miles.
Now, substitute this value of AC back into equation 1 to solve for h:
h = tan(20.5°) * AC ≈ tan(20.5°) * 1.099 miles
Calculate h using a calculator. The height of the balloon is approximately 0.394 miles, or about 2085 feet.