```Question 89021
{{{h = -16t^2 + 112t }}}

{{{180 = -16t^2 + 112t }}} Let h=180

{{{0 = -16t^2 + 112t-180 }}} Subtract 180 from both sides

Now let's use the quadratic formula to solve for t:

{{{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+112*t-180=0}}} ( notice {{{a=-16}}}, {{{b=112}}}, and {{{c=-180}}})

{{{t = (-112 +- sqrt( (112)^2-4*-16*-180 ))/(2*-16)}}} Plug in a=-16, b=112, and c=-180

{{{t = (-112 +- sqrt( 12544-4*-16*-180 ))/(2*-16)}}} Square 112 to get 12544

{{{t = (-112 +- sqrt( 12544+-11520 ))/(2*-16)}}} Multiply {{{-4*-180*-16}}} to get {{{-11520}}}

{{{t = (-112 +- sqrt( 1024 ))/(2*-16)}}} Combine like terms in the radicand (everything under the square root)

{{{t = (-112 +- 32)/(2*-16)}}} Simplify the square root

{{{t = (-112 +- 32)/-32}}} Multiply 2 and -16 to get -32

So now the expression breaks down into two parts

{{{t = (-112 + 32)/-32}}} or {{{t = (-112 - 32)/-32}}}

Lets look at the first part:

{{{x=(-112 + 32)/-32}}}

{{{t=-80/-32}}} Add the terms in the numerator

{{{t=5/2}}} Divide

{{{t=5/2}}}

Now lets look at the second part:

{{{x=(-112 - 32)/-32}}}

{{{t=-144/-32}}} Subtract the terms in the numerator

{{{t=9/2}}} Divide

{{{t=9/2}}}

So our solutions are:

{{{t=5/2}}} or {{{t=9/2}}}  (which are 2.5 and 4.5 in decimal form)

Notice when we graph {{{-16*x^2+112*x-180}}} (just replace t with x), we get:

{{{ graph( 500, 500, -5, 19, -5, 19,-16*x^2+112*x+-180) }}}

and we can see that the roots are {{{t=5/2}}} and {{{t=9/2}}}. This verifies our answer

So it takes 2.5 seconds to rise to the height of 180 feet, and 4.5 seconds to hit 180 feet.```