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.
Sure! I'd be happy to help you understand why the speed of the hot air balloon is equal to the square root of (10)^2 + (5)^2.
To find the speed of the balloon, we can use the Pythagorean theorem, which relates the lengths of the sides of a right triangle. In this case, the vertical speed of the balloon (10 ft/sec) and the horizontal speed of the wind (5 ft/sec) form the legs of a right triangle, and the speed of the balloon is the hypotenuse.
Let's denote the vertical speed as Vv (10 ft/sec) and the horizontal speed as Vh (5 ft/sec). According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, we have:
Speed of the balloon^2 = Vv^2 + Vh^2
Substituting the given values, we get:
Speed of the balloon^2 = 10^2 + 5^2
Simplifying further, we have:
Speed of the balloon^2 = 100 + 25
Speed of the balloon^2 = 125
To find the speed of the balloon, we need to take the square root of both sides of the equation. Taking the square root of 125 gives us:
Speed of the balloon = sqrt(125)
Since the square root of 125 is an irrational number, we can approximate it as:
Speed of the balloon ā 11.18 ft/sec
So, the speed of the balloon is approximately 11.18 ft/sec.
Do you have any more questions?