Question 200858
a)


{{{S(t)=-16t^2-32t+128}}} Start with the given function.



{{{S(1)=-16(1)^2-32(1)+128}}} Plug in {{{t=1}}}.



{{{S(1)=-16(1)-32(1)+128}}} Square {{{1}}} to get {{{1}}}.



{{{S(1)=-16-32(1)+128}}} Multiply {{{-16}}} and {{{1}}} to get {{{-16}}}.



{{{S(1)=-16-32+128}}} Multiply {{{-32}}} and {{{1}}} to get {{{-32}}}.



{{{S(1)=80}}} Combine like terms.



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



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b)


{{{S(t)=-16t^2-32t+128}}} Start with the given function.



{{{0=-16t^2-32t+128}}} Plug in {{{S(t)=0}}} (ie make the height equal to zero).



Notice that the quadratic {{{-16t^2-32t+128}}} is in the form of {{{At^2+Bt+C}}} where {{{A=-16}}}, {{{B=-32}}}, and {{{C=128}}}



Let's use the quadratic formula to solve for "t":



{{{t = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{t = (-(-32) +- sqrt( (-32)^2-4(-16)(128) ))/(2(-16))}}} Plug in  {{{A=-16}}}, {{{B=-32}}}, and {{{C=128}}}



{{{t = (32 +- sqrt( (-32)^2-4(-16)(128) ))/(2(-16))}}} Negate {{{-32}}} to get {{{32}}}. 



{{{t = (32 +- sqrt( 1024-4(-16)(128) ))/(2(-16))}}} Square {{{-32}}} to get {{{1024}}}. 



{{{t = (32 +- sqrt( 1024--8192 ))/(2(-16))}}} Multiply {{{4(-16)(128)}}} to get {{{-8192}}}



{{{t = (32 +- sqrt( 1024+8192 ))/(2(-16))}}} Rewrite {{{sqrt(1024--8192)}}} as {{{sqrt(1024+8192)}}}



{{{t = (32 +- sqrt( 9216 ))/(2(-16))}}} Add {{{1024}}} to {{{8192}}} to get {{{9216}}}



{{{t = (32 +- sqrt( 9216 ))/(-32)}}} Multiply {{{2}}} and {{{-16}}} to get {{{-32}}}. 



{{{t = (32 +- 96)/(-32)}}} Take the square root of {{{9216}}} to get {{{96}}}. 



{{{t = (32 + 96)/(-32)}}} or {{{t = (32 - 96)/(-32)}}} Break up the expression. 



{{{t = (128)/(-32)}}} or {{{t =  (-64)/(-32)}}} Combine like terms. 



{{{t = -4}}} or {{{t = 2}}} Simplify. 



So the <i>possible</i> 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.