A balloon is secured to rope that is staked to the ground. A breeze blows the balloon so that the rope is taut while the balloon is directly above a flag pole that is 30 feet from where the rope is staked down. Find the altitude of the balloon if the rope is 100 feet long. Leave your answer in simplified radical form.
30^2 + h^2 = 100^2
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To find the altitude of the balloon, we can use the Pythagorean theorem because we have a right triangle formed by the rope, the distance from the stake to the flagpole, and the altitude of the balloon.
According to the Pythagorean theorem, the square of the length of the hypotenuse (the rope) is equal to the sum of the squares of the other two sides (the distance from the stake to the flagpole and the altitude of the balloon).
Let's define the altitude of the balloon as 'x':
(sqrt(x^2 + 30^2))^2 = 100^2
Simplifying the equation:
x^2 + 900 = 10000
x^2 = 10000 - 900
x^2 = 9100
Now, we need to find the square root of both sides to solve for 'x':
sqrt(x^2) = sqrt(9100)
x = sqrt(9100)
Therefore, the altitude of the balloon is sqrt(9100) in simplified radical form.
To find the altitude 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 rope connecting the balloon to the ground forms the hypotenuse, while the distance between the rope and the flagpole is one of the other sides. Let's call the altitude of the balloon "h".
According to the problem, the rope is 100 feet long, and the distance between the rope and the flagpole is 30 feet. Using the Pythagorean theorem, we have:
h^2 + 30^2 = 100^2
Simplifying this equation, we get:
h^2 + 900 = 10,000
Subtracting 900 from both sides, we have:
h^2 = 9,100
Taking the square root of both sides, we find:
h = √9,100
Simplifying the square root, we get the final answer:
h = 2√2,275 feet
So, the altitude of the balloon is 2√2,275 feet.