a hot air balloon is tied to the ground with 200 ft. of rope. The wind is pushing the balloon to make a 23 degree angle with the ground. What is the real height of the balloon?
hypotenuse = 200
sin 23 = h/hypotenuse = h/200
so
h = 200 sin 23
= 78.1 ft
To find the real height of the balloon, we can use trigonometry. Let's consider the right triangle formed by the height of the balloon, the length of the rope, and the angle between the rope and the ground.
In this case, the length of the rope is the hypotenuse of the triangle. We have the angle between the rope and the ground, which is 23 degrees. We want to find the height of the balloon, which is the side opposite to the angle.
Using the sine function, we can set up the equation:
sin(23°) = height of the balloon / length of the rope
Now, let's solve for the height of the balloon:
height of the balloon = sin(23°) * length of the rope
height of the balloon = sin(23°) * 200 ft
To find the real height, we need to calculate the value of sin(23°) and then multiply it by 200 ft.
Using a scientific calculator or a math tool capable of calculating trigonometric functions, we find that sin(23°) is approximately 0.3917.
Now, we can substitute this value into the formula:
height of the balloon = 0.3917 * 200 ft
height of the balloon ≈ 78.34 ft
Therefore, the real height of the balloon is approximately 78.34 feet.