An object is projected directly upward from the ground. Its distance in feet from the ground in t seconds is s equals s=112t−16t^2
The object will be 160 feet from the ground after () second(s) and after
() second(s).
The object strikes the ground after () second(s).
16 t^2 -112 t +160 = 0
t^2 - 7 t + 10 = 0
(t-5)(t-2) = 0
so at 2 seconds on the way up and at 5 seconds on the way down
0 = 16t (7 - t)
crash at t = 7
To find the time it takes for the object to be 160 feet from the ground, we need to set the equation s=160 and solve for t.
s = 112t - 16t^2
160 = 112t - 16t^2
Rearranging the equation, we have:
16t^2 - 112t + 160 = 0
Dividing each term by 16:
t^2 - 7t + 10 = 0
Factoring the quadratic equation:
(t - 5)(t - 2) = 0
Setting each factor equal to zero:
t - 5 = 0 or t - 2 = 0
Solving for t:
t = 5 or t = 2
Therefore, the object will be 160 feet from the ground after 5 seconds and after 2 seconds.
To find the time it takes for the object to strike the ground, we need to set the equation s=0 and solve for t.
s = 112t - 16t^2
0 = 112t - 16t^2
16t^2 - 112t = 0
Dividing each term by 16:
t^2 - 7t = 0
Factoring out t:
t(t - 7) = 0
Setting each factor equal to zero:
t = 0 or t - 7 = 0
Solving for t:
t = 0 or t = 7
Therefore, the object will strike the ground after 7 seconds.
To find the time when the object will be 160 feet from the ground, we need to substitute the given value of 160 for s in the equation s = 112t - 16t^2 and solve for t.
112t - 16t^2 = 160
Rearranging the equation, we have -16t^2 + 112t - 160 = 0.
To solve this quadratic equation, we can either factor, complete the square, or use the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, the values of a, b, and c are -16, 112, and -160, respectively. Substituting these values into the quadratic formula:
t = (-(112) ± √((112)^2 - 4(-16)(-160))) / (2(-16))
Simplifying:
t = (-112 ± √(12544 - 10240)) / (-32)
t = (-112 ± √2304) / (-32)
t = (-112 ± 48) / (-32)
t = (-112 + 48) / (-32) or t = (-112 - 48) / (-32)
t = -64 / -32 or t = -160 / -32
t = 2 or t = 5
Therefore, the object will be 160 feet from the ground after 2 seconds and after 5 seconds.
To find the time when the object strikes the ground, we need to solve for t when s = 0.
112t - 16t^2 = 0
Factoring out t:
t(112 - 16t) = 0
Setting each factor equal to zero:
t = 0 (since time cannot be negative)
or
112 - 16t = 0
Solving for t:
16t = 112
t = 7
Therefore, the object strikes the ground after 7 seconds.