Colonel Harry a. Froboess of Switzerland jumped upward and outward from a windonw of a blimp and landed in a body of water on June 22, 1936. His jump was 120m, about the height of a 40-storey building. The path of his jump can be represented by the quadratic relationship h = 120 + 5.5t
- 4.9t^2,where h is the height above the water in metres and t is te time in seconds.
a) Find the maximum height of his dive.
b) How long did it take him to reach the water?
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I just want to know if when I am doing this equation, that the answer to b) is figuring out the x-intercept
To find the maximum height of his dive, we can use the formula h = 120 + 5.5t - 4.9t^2. In this equation, h represents the height above the water in meters, and t represents the time in seconds.
To find the maximum height, we need to determine the vertex of the quadratic equation. The vertex of a quadratic equation represented in standard form (ax^2 + bx + c) can be found using the formula t = -b / (2a).
In this case, a = -4.9 and b = 5.5. Plugging these values into the formula, we get:
t = -(5.5) / (2*(-4.9))
t = 5.5 / 9.8
t ≈ 0.561 seconds
To find the maximum height, we substitute this value of t into the equation:
h = 120 + 5.5 * 0.561 - 4.9 * (0.561)^2
h ≈ 120 + 3.086 - 1.943
h ≈ 121.143 meters
Therefore, the maximum height of Colonel Harry A. Froboess' dive was approximately 121.143 meters.
Now let's move on to determining how long it took him to reach the water.
To find the time it took him to reach the water, we need to determine when the height h equals zero. In other words, we need to find the values of t that make the quadratic equation equal to zero.
The formula h = 120 + 5.5t - 4.9t^2 becomes:
0 = 120 + 5.5t - 4.9t^2
To solve this equation, we can either factor it or use the quadratic formula. In this case, since the equation does not easily factor, we will use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a)
Given a = -4.9, b = 5.5, and c = -120, we can now substitute these values into the formula:
t = (-(5.5) ± √((5.5)^2 - 4 * (-4.9) * (-120))) / (2 * (-4.9))
Simplifying this expression, we get:
t = (-5.5 ± √(30.25 + 2352)) / (-9.8)
t = (-5.5 ± √(2382.25)) / (-9.8)
t = (-5.5 ± 48.8) / (-9.8)
We have two possible solutions using both the positive and negative square roots:
t_1 = (-5.5 + 48.8) / (-9.8) ≈ 4.47 seconds
t_2 = (-5.5 - 48.8) / (-9.8) ≈ -5.29 seconds
For a physical scenario like this, we choose the positive value since time cannot be negative.
Therefore, it took approximately 4.47 seconds for Colonel Harry A. Froboess to reach the water.