If a person throws a water balloon, the height (in feet) if the water balloon is modeled by the equation: h=-16t^2-5t+100, where t is in seconds. When will the water balloon hit the ground?
set the equation to 0 and solve for t. you may have to use the quadratic formula...
To find the time when the water balloon hits the ground, we need to determine when the height (h) is equal to zero.
The equation given is: h = -16t^2 - 5t + 100
Setting h to zero: 0 = -16t^2 - 5t + 100
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's solve it using the quadratic formula:
The quadratic formula states that, for an equation of the form: ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case: a = -16, b = -5, and c = 100.
Applying the quadratic formula:
t = (-(-5) ± √((-5)^2 - 4(-16)(100))) / (2(-16))
Simplifying the equation:
t = (5 ± √(25 + 6400)) / (-32)
t = (5 ± √(6425)) / (-32)
Now, calculating the value inside the square root:
t = (5 ± √(6425)) / (-32)
t ≈ (5 ± 80.156) / (-32)
Simplifying further:
t ≈ (5 + 80.156) / (-32) ≈ 75.156 / (-32) ≈ -2.348
or
t ≈ (5 - 80.156) / (-32) ≈ -75.156 / (-32) ≈ 2.35
Since time cannot be negative, we discard the negative value.
Therefore, the water balloon will hit the ground approximately 2.35 seconds after it is thrown.
To find when the water balloon hits the ground, we need to determine the value of t that makes the height (h) zero. In other words, we need to solve the equation -16t^2 - 5t + 100 = 0.
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula.
The quadratic formula states that for an equation ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = -16, b = -5, and c = 100. Plugging these values into the quadratic formula, we get:
t = (-(-5) ± sqrt((-5)^2 - 4(-16)(100))) / (2(-16))
t = (5 ± sqrt(25 + 6400)) / (-32)
t = (5 ± sqrt(6425)) / (-32)
Now, we need to simplify the square root of 6425. By using a calculator or an online tool, we find that sqrt(6425) ≈ 80.156.
Substituting this value back into the equation, we have:
t = (5 ± 80.156) / (-32)
To find the two possible values of t, we have two cases:
Case 1: t = (5 + 80.156) / (-32)
t ≈ -2.170
Case 2: t = (5 - 80.156) / (-32)
t ≈ 2.450
Since time cannot be negative in this context, we discard the negative value and conclude that the water balloon hits the ground approximately 2.450 seconds after it is thrown.