If an object is propelled upward from a height of s feet at an initial velocity of v feet per second, then its height h after t seconds is given by the equation hequals
minus16t squaredplusvtpluss,
where h is in feet. If the object is propelled from a height of 12
feet with an initial velocity of 96
feet per second, its height h is given by the equation hequals
minus16t squaredplus96tplus12
.
After how many seconds is the height 120
feet?
The time is nothing
seconds.
120 = -16 t^2 + 96 t + 12
0 = -4 t^2 + 24 t - 27
use quadratic formula to find t
To find the time when the height is 120 feet, we can set the height equation to 120 and solve for t.
Given height equation: h = -16t^2 + 96t + 12
Setting h = 120:
120 = -16t^2 + 96t + 12
Rearranging the equation:
16t^2 - 96t + 12 - 120 = 0
Simplifying:
16t^2 - 96t - 108 = 0
To solve this quadratic equation, we can factor it:
4(4t^2 - 24t - 27) = 0
Setting each factor to zero:
4t^2 - 24t - 27 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case:
a = 4
b = -24
c = -27
Plugging these values into the quadratic formula:
t = (-(-24) ± √((-24)^2 - 4(4)(-27))) / (2(4))
t = (24 ± √(576 + 432)) / 8
t = (24 ± √(1008)) / 8
t = (24 ± √(16 * 63)) / 8
t = (24 ± 4√(63)) / 8
t = (3 ± √(63)) / 2
So, the solutions for t are:
t1 = (3 + √(63)) / 2
t2 = (3 - √(63)) / 2
Therefore, the height of 120 feet is reached at t1 = (3 + √(63)) / 2 seconds and t2 = (3 - √(63)) / 2 seconds.
To find the time at which the height is 120 feet, we need to solve the equation -16t^2 + 96t + 12 = 120.
1. Substitute h = 120 into the equation: -16t^2 + 96t + 12 = 120.
2. Subtract 120 from both sides of the equation to isolate the quadratic term: -16t^2 + 96t - 108 = 0.
3. Divide the entire equation by -4 to simplify the equation: 4t^2 - 24t + 27 = 0.
At this point, we have a quadratic equation. We can solve it using factoring, completing the square, or the quadratic formula. In this case, the quadratic equation can be factored:
4t^2 - 24t + 27 = 0
(2t - 3)(2t - 9) = 0
Using the zero-product property, we set each factor equal to zero and solve for t:
2t - 3 = 0 --> 2t = 3 --> t = 3/2 or 1.5 seconds
2t - 9 = 0 --> 2t = 9 --> t = 9/2 or 4.5 seconds
Therefore, the object will reach a height of 120 feet after 1.5 seconds and 4.5 seconds.