Answer
a. $100 ft$
b. $16\frac{ft}{s}$ and $-16\frac{ft}{s}$
Work Step by Step
a. $v(t) = \frac{d(s)}{dx}$
$v'(t) = \frac{d(80t)}{dt} - \frac{16t^{2}}{dt}$
$v'(t) = 80 - 32t$
$v'(t) = 0$
$0 = 80 - 32t$
Solve for $t$:
$32t = 80$
$t = \frac{80}{32}$
$t = 2.5 s$
$s = 80t - 16t^{2}$
$s = 80(2.5) - 16(2.5)^2$
$s = 200 - 16(6.25)$
$s = 200 - 100$
$s = 100 ft$
b. $s = 80t - 16t^{2}$
$96 = 80t -16t^{2}$
$80t -16t^{2}-96 = 0$
Simplify by $-16$:
$-16(t^{2} - 5t +6) = 0$
$t^{2} - 5t + 6 = 0$
Now factorize:
$(t-3)(t-2) = 0$
$t = 3$ and $t=2$
$v(t) = \frac{ds}{dt}$
$v(t) = 80 - 32t$
$v(2) = 80 - 32(2)$
$v(2) = 80 - 64$
$v(2) = 16\frac{ft}{s}$
$v(3) = 80 - 32(3)$
$v(3) = 80 - 96$
$v(3) = -16\frac{ft}{s}$
