A projectile is launched directly upward from the top of a 112 - foot building with an initial velocity (speed) of 96 feet per second. This translates to the h(t) = -16t^2 + 96t + 112 where h(t) is the height of the projectile in feet and t is the time in seconds after it is launched. When will the projectile hit the ground?
t = 7 seconds after it is launched
Is this correct?
16 t^2 - 96 t - 112 = 0
t^2 - 6 t - 7 = 0
(t-7)(t+1) = 0
t = 7
so yes
Thanks Damon!
To find when the projectile hits the ground, we need to determine the time when the height of the projectile is equal to zero.
Given the equation h(t) = -16t^2 + 96t + 112, we set h(t) equal to zero and solve for t:
-16t^2 + 96t + 112 = 0
To solve this quadratic equation, we can either factor, use the quadratic formula, or complete the square. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In the given equation, a = -16, b = 96, and c = 112. Plugging these values into the quadratic formula, we get:
t = (-96 ± √(96^2 - 4(-16)(112))) / (2(-16))
Calculating further:
t = (-96 ± √(9216 + 7168))/ (-32)
t = (-96 ± √(16384))/ (-32)
t = (-96 ± 128)/ (-32)
Now let's find the two possible values for t:
t₁ = (-96 + 128)/ (-32)
t₁ = 32/ (-32)
t₁ = -1
t₂ = (-96 - 128)/ (-32)
t₂ = -224/ (-32)
t₂ = 7
From the calculations, we find that t = -1 and t = 7. However, time cannot be negative in this context. Therefore, the correct answer is t = 7 seconds.
Yes, your answer is correct. The projectile will hit the ground 7 seconds after it is launched.
To find the time at which the projectile hits the ground, we need to solve for t when h(t) = 0.
The equation for the height of the projectile is given by h(t) = -16t^2 + 96t + 112.
Setting h(t) to 0, we have:
0 = -16t^2 + 96t + 112
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula.
Using the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by x = (-b ± sqrt(b^2 - 4ac)) / (2a), we can find the values of t when h(t) = 0.
In this case, a = -16, b = 96, and c = 112. Substituting these values into the quadratic formula, we have:
t = (-96 ± sqrt(96^2 - 4(-16)(112))) / (2*(-16))
Simplifying this expression further, we have:
t = (-96 ± sqrt(9216 + 7168)) / (-32)
t = (-96 ± sqrt(16384)) / (-32)
t = (-96 ± 128) / (-32)
Now, to find the two possible values of t:
1. t = (-96 + 128) / -32 = 32 / -32 = -1
2. t = (-96 - 128) / -32 = -224 / -32 = 7
From the calculation, we find that t can be -1 or 7.
However, time cannot be negative in this context, so the correct answer is t = 7 seconds, which means the projectile will hit the ground 7 seconds after it is launched.
Therefore, your answer of t = 7 seconds is correct.