Question 180814
a)

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



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


---------------------------------------------------

b)


The wrench will reach the ground when {{{S(t)=0}}} (ie the height of the wrench is zero)



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



{{{0=-16t^2-32t+128}}} Plug in {{{S(t)=0}}}




Notice we have a quadratic equation 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 possible answers are {{{t = -4}}} or {{{t = 2}}} 

  

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



============================================================


Answer:


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