SOLUTION: Throwing a wrench. 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 abo

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Question 180814: Throwing a wrench. 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?
Found 2 solutions by jim_thompson5910, eperette:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)
S%28t%29=-16t%5E2-32t%2B128 Start with the given equation.


S%281%29=-16%281%29%5E2-32%281%29%2B128 Plug in t=1.


S%281%29=-16%281%29-32%281%29%2B128 Square 1 to get 1.


S%281%29=-16-32%281%29%2B128 Multiply -16 and 1 to get -16.


S%281%29=-16-32%2B128 Multiply -32 and 1 to get -32.


S%281%29=80 Combine like terms.


So after one second, the wrench is 80 ft above the ground.

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b)

The wrench will reach the ground when S%28t%29=0 (ie the height of the wrench is zero)


S%28t%29=-16t%5E2-32t%2B128 Start with the given equation.


0=-16t%5E2-32t%2B128 Plug in S%28t%29=0



Notice we have a quadratic equation in the form of at%5E2%2Bbt%2Bc where a=-16, b=-32, and c=128


Let's use the quadratic formula to solve for t


t+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


t+=+%28-%28-32%29+%2B-+sqrt%28+%28-32%29%5E2-4%28-16%29%28128%29+%29%29%2F%282%28-16%29%29 Plug in a=-16, b=-32, and c=128


t+=+%2832+%2B-+sqrt%28+%28-32%29%5E2-4%28-16%29%28128%29+%29%29%2F%282%28-16%29%29 Negate -32 to get 32.


t+=+%2832+%2B-+sqrt%28+1024-4%28-16%29%28128%29+%29%29%2F%282%28-16%29%29 Square -32 to get 1024.


t+=+%2832+%2B-+sqrt%28+1024--8192+%29%29%2F%282%28-16%29%29 Multiply 4%28-16%29%28128%29 to get -8192


t+=+%2832+%2B-+sqrt%28+1024%2B8192+%29%29%2F%282%28-16%29%29 Rewrite sqrt%281024--8192%29 as sqrt%281024%2B8192%29


t+=+%2832+%2B-+sqrt%28+9216+%29%29%2F%282%28-16%29%29 Add 1024 to 8192 to get 9216


t+=+%2832+%2B-+sqrt%28+9216+%29%29%2F%28-32%29 Multiply 2 and -16 to get -32.


t+=+%2832+%2B-+96%29%2F%28-32%29 Take the square root of 9216 to get 96.


t+=+%2832+%2B+96%29%2F%28-32%29 or t+=+%2832+-+96%29%2F%28-32%29 Break up the expression.


t+=+%28128%29%2F%28-32%29 or t+=++%28-64%29%2F%28-32%29 Combine like terms.


t+=+-4 or t+=+2 Simplify.


So the possible answers are t+=+-4 or t+=+2

However, you CANNOT have a negative time. So the only answer is t+=+2


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Answer:

So it takes 2 seconds for the wrench to hit the ground

Answer by eperette(173) About Me  (Show Source):
You can put this solution on YOUR website!
S(t)= -16t^2 -32t + 128
(a) need to find S(t) when t=1
S(1)= -16(1)^2 - 32(1) + 128
S(1)= -16 -32 + 128
S(1)= 80
At t=1second the height of the wrench is 80ft
(b)if the wrench reaches the ground, then the height is 0, ...in other words it is asking you to find t when S(t)=0....you can solve this problem two ways
(Graphing Method)
With your calculator, graph the equation...your solution will be your zeros (or x-intercepts)....notice that x-intercepts happen at t=-4 and t=2....t=-4 does not make sense so your answer should be t=2 seconds
(Algebra Method)
0 = -16t^2 - 32t +128
0 = t^2 -32t -2048
0 = (t-64)(t+32)
0 = (t-64/-16)(t + 32/-16)
0 = (t + 4)(t-2)
(t+4)=0 or (t-2)=0
t=-4 or t=2
it would not make sense of the time to be negative....so this leaves us with one answer: the wrench reaches the ground when t=2 seconds