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?
4 answers
The height of the balloon can be calculated using the formula h = d*tan(a), where h is the height of the balloon, d is the distance between the two friends, and a is the angle of elevation. In this case, h = 1.0*tan(20.5°) = 0.37 miles.
AAAaannndd the bot gets it wrong yet again!
h cot20.5° - h cot25.5° = 1
h = 1.73 mi
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
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