Question 276918
Let's see if we can "un"perplex you!
Given:
{{{S(t) = -16t^2-32+128}}} This is the quadratic equation that shows the relationship of the height (h or S) of an object as a function of time (t).
The general form is:
{{{h(t) = -(1/2)gt^2 +v[0]t+h[0]}}} where h (or S) = height of object, {{{v[0]}}}= initial velocity, a negative {{{v[0]}}} means object is going down, and {{{h[0]}}}is the initial height of the object, so, to find the height of the object after 1 second, substitute t = 1...
{{{S(t) = -16t^2-32t+128}}} Substitute t=1
{{{S(1) = -16(1)^2-32(1)+128}}}
{{{S(1) = -48+128}}}
{{{S(1) = 80}}}feet.
The hammer (or was it a wrench?) would be 80 feet above the ground after 1 second had elapsed.
The second question...How long does it take for it (the wrench) to reach the ground? So you are really asking..."At what time t will the height S be zero?"
{{{S(t) = -16t^2-32t+128}}} Set {{{S(t) = 0}}} and solve for t.
{{{-16t^2-32t+128 = 0}}} First factor (-16) to simplify the equation a bit.
{{{-16(t^2+2t-8) = 0}}} so, from the zero product rule, we get...
{{{t^2+2t-8 = 0}}} Now factor this trinomial.
{{{(t+4)(t-2) = 0}}} which means that...
{{{t+4 = 0}}} or {{{t+2 = 0}}} from which we get...
{{{t = -4}}} or {{{highlight(t = 2)}}} Discard the negative solution.
So it would take 2 seconds for the wrench to reach the ground.
P.S. Your statement "The height of the wrench above the ground after 6 seconds is given by {{{S(t) = -16t^2-32T+128}}}" does not make sense.  The equation is ok but this will tell you the height above ground after t seconds.
To find the height above ground after 6 seconds, you would have to substitute t = 6 into the equation and solve for S(6).