Question 92401
"a. If the arrow reaches its maximum height in 4 seconds, find its maximum height"



Lets find {{{h}}} when {{{t=4}}}


{{{h=-16t^2+128t+144}}} Start with the given polynomial



{{{h=-16(4)^2+128(4)+144}}} Plug in {{{t=4}}}



{{{h=-16(16)+128(4)+144}}} Raise 4 to the second power to get 16



{{{h=-256+128(4)+144}}} Multiply -16 by 16 to get -256



{{{h=-256+512+144}}} Multiply 128 by 4 to get 512



{{{h=400}}} Now combine like terms



So when {{{t=4}}}, {{{h=400}}}


So the maximum height is 400 feet


<hr>


"b. How many seconds will it take for the arrow to reach the ground?"


Let h=0 and solve for t


{{{0=-16t^2+128t+144}}}




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



Starting with the general quadratic


{{{at^2+bt+c=0}}}


the general solution using the quadratic equation is:


{{{t = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{-16*t^2+128*t+144=0}}} ( notice {{{a=-16}}}, {{{b=128}}}, and {{{c=144}}})


{{{t = (-128 +- sqrt( (128)^2-4*-16*144 ))/(2*-16)}}} Plug in a=-16, b=128, and c=144




{{{t = (-128 +- sqrt( 16384-4*-16*144 ))/(2*-16)}}} Square 128 to get 16384  




{{{t = (-128 +- sqrt( 16384+9216 ))/(2*-16)}}} Multiply {{{-4*144*-16}}} to get {{{9216}}}




{{{t = (-128 +- sqrt( 25600 ))/(2*-16)}}} Combine like terms in the radicand (everything under the square root)




{{{t = (-128 +- 160)/(2*-16)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{t = (-128 +- 160)/-32}}} Multiply 2 and -16 to get -32


So now the expression breaks down into two parts


{{{t = (-128 + 160)/-32}}} or {{{t = (-128 - 160)/-32}}}


Lets look at the first part:


{{{x=(-128 + 160)/-32}}}


{{{t=32/-32}}} Add the terms in the numerator

{{{t=-1}}} Divide


So one answer is

{{{t=-1}}}




Now lets look at the second part:


{{{x=(-128 - 160)/-32}}}


{{{t=-288/-32}}} Subtract the terms in the numerator

{{{t=9}}} Divide


So another answer is

{{{t=9}}}


So our possible solutions are:

{{{t=-1}}} or {{{t=9}}}


However, a negative time doesn't make sense. So our only answer is 


{{{t=9}}}


So it takes 9 seconds to reach the ground