Math, Please Help!!!
posted by BOB .
a hot air balloon is rising vertically 10 ft/sec while the wind is blowing horizontally 5ft/sec. Find the speed of the balloon and the angle x that it makes with the horizontal.
I know that the speed of the balloon is
the square root of (10)^2 + (5)^2, but I was wondering if someone could tell me why the speed is equal to that. Thanks.

from the origin draw a vertical line of length 10
from the origin draw a horizontal line of length 5
Complete the rectangle, and draw in the diagonal.
That diagonal is the resultant velocity.
You now have a rightangled triangle with base of 5 and height of 10
let the hypotenuse be h
h^2 = 10^2+5^2 = 125
h = √125 = 5√5 = appr. 11.18
to find the angle
sinx = 10/11.18
x = 63.4°
we could have done : tanx = 10/5 = 2
x = 63.4° 
You can draw a triangle for velocity, even though usually you do it for distances, because the distance is velocity * time. The time elapsed is just a scale factor, so the velocity triangle is just the distance triangle, scaled by how much time has elapsed.