A ball is thrown downward from the top of a 100-foot building with an initial velocity of 14 feet per second. The height of the ball h after t seconds is given by the equation h=-16t^2-14t+100.
How long after the ball is thrown will it strike the ground?
To find out how long after the ball is thrown will it strike the ground, we need to determine the time when the height of the ball (h) becomes zero.
Given the equation h = -16t^2 - 14t + 100, we can set h = 0 and solve for t.
So, -16t^2 - 14t + 100 = 0
To solve this quadratic equation, we can use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / 2a, where a = -16, b = -14, and c = 100.
Plugging in these values, we get:
t = (-(-14) ± √((-14)^2 - 4(-16)(100))) / (2(-16))
Simplifying further:
t = (14 ± √(196 + 6400)) / (-32)
t = (14 ± √(6596)) / (-32)
Now, we can calculate the two possible values of t:
t1 = (14 + √(6596)) / (-32)
t2 = (14 - √(6596)) / (-32)
Finding the square root of 6596 gives us:
t1 ≈ -0.792
t2 ≈ 5.292
Since time cannot be negative in this context, we only consider the positive value of t, which is approximately 5.292 seconds.
Therefore, the ball will strike the ground approximately 5.292 seconds after it is thrown.
To find the time it takes for the ball to strike the ground, we need to find the value of t when the height h is equal to zero.
Given equation: h = -16t^2 - 14t + 100
Setting h to zero: 0 = -16t^2 - 14t + 100
Now, we have a quadratic equation that we can solve to find the value of t.
To solve this equation, we can use the quadratic formula which states that for an equation in the form ax^2 + bx + c = 0, the solution for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -16, b = -14, and c = 100. Plugging these values into the quadratic formula, we get:
t = (-(-14) ± √((-14)^2 - 4(-16)(100))) / (2(-16))
Simplifying further:
t = (14 ± √(196 + 6400)) / (-32)
t = (14 ± √(6596)) / (-32)
Now, let's calculate the discriminant inside the square root:
D = 6596
The discriminant is positive, which means we will have two real solutions.
Using a calculator, we can find the square root of 6596:
√(6596) ≈ 81.16
Now, substituting the value of the discriminant into our equation:
t = (14 ± 81.16) / (-32)
We have two possibilities:
t₁ = (14 + 81.16) / (-32)
t₁ ≈ -2.517
t₂ = (14 - 81.16) / (-32)
t₂ ≈ 2.816
We discard the negative value (t₁) for time since it would not make sense in this context. Therefore, the ball will strike the ground approximately 2.816 seconds after it is thrown.
Garbage
It hits the ground when the height is back to zero.
just solve for t when h=0
use your normal methods to solve
16t^2 + 14t - 100 = 0