An angry construction worker throws his wrench downward from a height of 128 feet with an
initial velocity of 32 feet per second. The height of the wrench above the ground after t seconds is given
by S(t)= -16t^2 - 32t + 128.
a) What is the height of the wrench after 1 second?
b) How long does it take for the wrench to reach the ground?
a) replace t with 1
b) solve
-16t^2 - 32t + 128 = 0
or , after dividing all terms by -16,
t^2 + 2t - 8 = 0
a) To find the height of the wrench after 1 second, we need to substitute t = 1 into the equation S(t) = -16t^2 - 32t + 128.
S(1) = -16(1)^2 - 32(1) + 128
= -16 - 32 + 128
= 80
Therefore, the height of the wrench after 1 second is 80 feet.
b) To determine how long it takes for the wrench to reach the ground, we need to find the value of t when S(t) = 0. This indicates that the height of the wrench is at ground level.
So, we need to solve the equation -16t^2 - 32t + 128 = 0.
Let's first divide the equation by -16 to simplify it:
t^2 + 2t - 8 = 0
Now, we can solve this quadratic equation. We can either use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 2, and c = -8.
Applying the quadratic formula:
t = (-2 ± √(2^2 - 4(1)(-8))) / (2(1))
= (-2 ± √(4 + 32)) / 2
= (-2 ± √36) / 2
= (-2 ± 6) / 2
We have two solutions:
t1 = (-2 - 6) / 2 = -8/2 = -4
t2 = (-2 + 6) / 2 = 4/2 = 2
Since time cannot be negative in this context, we ignore the negative value, t1.
Therefore, it takes 2 seconds for the wrench to reach the ground.