Why hovering near the top of a waterfall of a national park of 6400 feet, a helicopter pilot drops his sunglasses. The height h(t) of the sunglasses after t seconds is given by the polynomial function h(t) = -16t(squared) + 6400. When will the sunglasses hit the ground? (in seconds). Thank you.
when is -16t^2 + 6400 = 0 ?
16t^2 = 6400
t^2 = 400
t = 20
To find the time it takes for the sunglasses to hit the ground, we need to set h(t) equal to 0 and solve for t.
Given that h(t) = -16t^2 + 6400, we set h(t) = 0:
0 = -16t^2 + 6400
Now, to solve for t, we can divide both sides of the equation by -16:
0 = t^2 - 400
Next, rearrange the equation to isolate t^2:
t^2 = 400
Taking the square root of both sides, we get:
t = ±√400
Since time cannot be negative, we can ignore the negative square root, leaving us with:
t = √400
Simplifying, t = 20
Therefore, the sunglasses will hit the ground after 20 seconds.
To find when the sunglasses will hit the ground, we need to determine the value of t when h(t) equals zero.
The given polynomial function for the height of the sunglasses is h(t) = -16t^2 + 6400.
Since we want to find when the height is zero, we can set h(t) equal to zero and solve for t:
-16t^2 + 6400 = 0
To solve this quadratic equation, we can apply the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation in the form of ax^2 + bx + c = 0.
In this case, a = -16, b = 0, and c = 6400. Substituting these values into the quadratic formula:
t = (-0 ± √(0^2 - 4(-16)(6400))) / (2(-16))
t = (± √(0 - (-409600))) / (-32)
t = (± √(409600)) / (-32)
t = (± 640) / (-32)
We can simplify this further by dividing both the numerator and denominator by 32:
t = ± 20
Therefore, we have two possible solutions: t = 20 or t = -20.
However, since time cannot be negative in this context, we discard the negative solution. Thus, the sunglasses will hit the ground after 20 seconds.