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.
Every second the balloon rises 10 ft and moves sideways 5 feet.
So, the distance traveled in one second is sqrt(10^2 + 5^2)
Since it traveled that distance in one second, and speed is distance/time, the same triangle can be used to calculate the speed.
To find the speed of the balloon, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the vertical speed of the balloon is 10 ft/sec, and the horizontal speed due to the wind is 5 ft/sec. Let's call the speed of the balloon "v" and write out the equation using the Pythagorean theorem:
v^2 = (vertical speed)^2 + (horizontal speed)^2
Substituting the given values:
v^2 = 10^2 + 5^2
v^2 = 100 + 25
v^2 = 125
To find the value of v, we take the square root of both sides:
v = √125
Now, simplify the square root of 125:
v ≈ √(25 × 5)
v ≈ √25 × √5
v ≈ 5√5
So, the speed of the balloon is approximately 5√5 ft/sec.
To find the angle x that the balloon makes with the horizon, we can use trigonometry. The horizontal speed of the balloon is adjacent to the angle x, and the vertical speed is the opposite side.
Recall the trigonometric ratio for tangent:
tan(x) = (opposite) / (adjacent)
In this case, the vertical speed is 10 ft/sec (opposite) and the horizontal speed is 5 ft/sec (adjacent). Let's rearrange the equation to solve for x:
tan(x) = 10 / 5
tan(x) = 2
To find the value of x, we take the inverse tangent (or arctan) of both sides:
x = arctan(2)
Using a calculator, we find that x is approximately 63.43 degrees.
Therefore, the speed of the balloon is about 5√5 ft/sec, and the angle x it makes with the horizontal is approximately 63.43 degrees.