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 200858This question is from textbook Elementary and Intermediate Algebra
: 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) = -16t2 - 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?
 This question is from textbook Elementary and Intermediate Algebra

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 function.


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 the wrench is 80 feet in the air after 1 second.


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b)

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


0=-16t%5E2-32t%2B128 Plug in S%28t%29=0 (ie make the height equal to zero).


Notice that the quadratic -16t%5E2-32t%2B128 is 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 solutions are t+=+-4 or t+=+2

But remember that a negative time value doesn't make much sense. So we'll ignore t+=+-4


So the only solution is t+=+2 which means that it takes 2 seconds for the wrench to hit the ground.